Enter your Sign on user name and password.

Forgot password?
Sign In | Subscribe
Start learning today, and be successful in your academic & professional career. Start Today!
Use Chrome browser to play professor video
Vincent Selhorst-Jones

Vincent Selhorst-Jones

Using Matrices to Solve Systems of Linear Equations

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
II. Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
III. Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
VII. Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
VIII. Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
IX. Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
XI. Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
XII. Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
XIII. Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
XIV. Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
Loading...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Pre Calculus
  • Discussion

  • Study Guides

  • Practice Questions

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
Get immediate access to our entire library.

Sign up for Educator.com

Membership Overview

  • Unlimited access to our entire library of courses.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lesson files for programming and software training practice.
  • Track your course viewing progress.
  • Download lecture slides for taking notes.
  • Learn at your own pace... anytime, anywhere!

Using Matrices to Solve Systems of Linear Equations


    Note: The ideas in this lesson can be rather difficult to follow with just words. The video will help explain this a lot, as it has a lot of visual diagrams to show what's going on step-by-step.
  • We can represent an entire linear system with an an augmented matrix:
    x
    + y
    + z
    =
    3
    2x
    −2y
    +3z
    =
    −4
    −x
    −z
    =
    0
        ⇒    



    1
    1
    1
    3
    2
    −2
    3
    −4
    −1
    0
    −1
    0










     
    • Each row represents an equation of the system,
    • Each left-side column gives a variable's coefficients,
    • If a variable does not appear, it has coefficient 0,
    • The vertical line represents `=',
    • The right-side column gives the constant terms.
  • There are three row operations that we can perform on an augmented matrix:
    1. Interchange the locations of two rows.
    2. Multiply (or divide) a row by a nonzero number.
    3. Add (or subtract) a multiple of one row to another.
    While row operations are a simple idea, working with them involves a lot of arithmetic and steps. With that much calculation going on, it's easy to make mistakes. Counter this by noting each step you take and what you did. This will make it easier to avoid making mistakes and will help you find any that manage to creep through.
  • We can use row operations to put a matrix in reduced row-echelon form. While it has a formal definition, it will be enough for us to think of it as a diagonal of 1's starting at the top-left with 0's above and below (entries on the far right can be any number). If we can use row operations to put an augmented matrix in reduced row-echelon form, we will have solved its associated linear system.
  • Gauss-Jordan elimination is a method we can follow to produce reduced row-echelon form matrices through row operations and thus solve linear systems.
    1. Write the linear system as an augmented matrix.
    2. Use row operations to attain a 1 in the top left and zeros below. Then move on to creating the next 1 diagonally down, with zeros below. Repeat until end.
    3. Now, work from the bottom right of the diagonal, canceling out everything above the 1's. Continue up the diagonal until you have only 0's above as well.
    4. The matrix should now be in reduced row-echelon form, giving you the solutions to the system. [Note: If there is no solution or infinitely many, you will not be able to achieve reduced row-echelon form.]
  • We can also find the solutions to a linear system by using determinants and Cramer's Rule. Let A be the coefficient matrix for our linear system. [The coefficient matrix is the left-hand side of an augmented matrix. It is all the coefficients from the linear system arranged in a matrix.] Let Ai be the same as A except the ith column is replaced with the column of constants from the linear system (the right side of the equations). Then, if det(A) ≠ 0, the ith variable xi is
    xi =

    det
     
    Ai


    det
     
    A



     
    .
    If det(A) = 0, then the system has either no solutions or infinitely many solutions.
  • Using matrix multiplication, we can write a linear system as an equation with matrices:
           AX = B,    where
    • A is the coefficient matrix of the linear system,
    • X is a single-column matrix of the variables,
    • B is a single-column matrix of the constants.
    Notice that if we could get X alone on one side of the equation, we will have solved for the variables.
  • If A is invertible, then there exists some A−1 we can multiply by that will cancel out A above, allowing us to get X alone. By computing A−1 and A−1B, we will have solved the system. (Notice that we have to multiply A−1 from the left on both sides of the equation because matrix multiplication is affected by direction.) If A is not invertible (det(A) = 0), then the system either has no solutions or infinitely many solutions.
  • While all of these methods work great for solving linear systems, they all have the downside of being tedious: they take lots and lots of arithmetic. Good news! Almost all graphing calculators have the ability to do matrix (and vector) operations. You can enter matrices, then multiply them, take determinants, find inverses, or put them in reduced-row echelon form. Even if you don't have a graphing calculator, there are many websites where you can do these things for free.

Using Matrices to Solve Systems of Linear Equations

Convert the below linear system into an augmented matrix.
3x
+ 5y
=
−7
4x
−3y
=
10
  • An augmented matrix represents an entire linear system. The left side of the matrix has an entry for every coefficient attached to the variables, while the right side gives the constant terms.
  • As long as the order of variables is the same for each equation, we can just pull out each number based on its location, then slot them into a matrix:



    3
    5
    −7
    4
    −3
    10



[
3
5
−7
4
−3
10
]
Convert the below linear system into an augmented matrix.
3a+2b−4c=6,        5d−a=13,        b+c+d=0,        12a+3b−5d=3
  • An augmented matrix represents an entire linear system. The left side of the matrix has an entry for every coefficient attached to the variables, while the right side gives the constant terms.
  • To convert a linear system into an augmented matrix, each of the variables must appear in the same order for each of the equations. This includes variables that do not appear in a given equation. Since they do not appear in the equation, they have a 0 in front of them (because none of that variable is there). Put the linear system into such an order. Let's put the variables into the order a, b, c, d. This is not the only possible choice, but it seems natural since it's an alphabetic ordering.
    3a
    +2b
    −4c
    +0d
    =
    6
    −1a
    +0b
    +0c
    +5d
    =
    13
    0a
    +1b
    +1c
    +1d
    =
    0
    12a
    +3b
    +0c
    −5d
    =
    3
    [Note: This is not the only possible ordering for the linear system. We could have put each equation into any row. We put them top-to-bottom in order of left-to-right appearance in the problem, but any vertical ordering would have been acceptable as long as we didn't split up the equations. Similarly, any ordering of the variables within the equations would be fine. There are many possible ways to convert the linear system into an augmented matrix, this is just (probably) the most convenient.]
  • Once the linear system is properly ordered and it is clear what the coefficient to each variable is, we can just pull out each number based on its location, then slot them into a matrix:






    3
    2
    −4
    0
    6
    −1
    0
    0
    5
    13
    0
    1
    1
    1
    0
    12
    3
    0
    −5
    3






[
3
2
−4
0
6
−1
0
0
5
13
0
1
1
1
0
12
3
0
−5
3
] [Note: There are many possible answers because there are many ways to order the linear system before converting it into an augmented matrix. The above is likely the most convenient way, but it is not the only one. If you are interested in a little more discussion of this, see the second step to this question.]
Using Gauss-Jordan elimination, solve the linear system below.
4x
+ 7y
=
6
x
+3y
=
−1
  • The method of Gauss-Jordan elimination allows us to solve a linear system through augmented matrices and row operations. Begin by turning the linear system into an augmented matrix:



    4
    7
    6
    1
    3
    −1



  • Row operations allow us to do the following three things: add rows together, multiply an entire row by a number, and swap the location of two rows. Often we will combine the first two operations into a single action. Through the use of row operations, we turn the left half of the augmented matrix into an identity matrix. Gauss-Jordan elimination tells us to begin by making the main diagonal nothing but 1's, and everything below the diagonal 0's. For this problem, we can get a 1 in the top-left simply by swapping the rows, so we do that first:



    4
    7
    6
    1
    3
    −1



    R2 ↔ R1
    R2 ↔ R1



    1
    3
    −1
    4
    7
    6



    Then turn the number below the top-left 1 to a 0:
    −4R1 +R2



    1
    3
    −1
    0
    −5
    10



    Finally, we turn the bottom-right number into a 1 as well to complete the main diagonal:
    1

    5
    R2



    1
    3
    −1
    0
    1
    −2



  • Now that the main diagonal is 1's and there are 0's below, we reverse and work back up the main diagonal, turning all the numbers above the main diagonal to 0's as well.
                              
    −3R2+R1



    1
    0
    5
    0
    1
    −2



    At this point the Gauss-Jordan elimination process has given us a matrix in reduced row-echelon form, and we can turn the augmented matrix back into a linear system:
    1x
    +0y
    =
    5
    0x
    +1y
    =
    −2
    This clearly gives us x=5 and y=−2.
x=5,   y=−2
Using Gauss-Jordan elimination, solve the linear system below.
a
+ b
−2c
=
3
2a
+ 3b
−c
=
2
−a
+ 2b
+4c
=
−15
  • The method of Gauss-Jordan elimination allows us to solve a linear system through augmented matrices and row operations. Begin by turning the linear system into an augmented matrix:




    1
    1
    −2
    3
    2
    3
    −1
    2
    −1
    2
    4
    −15




  • Row operations allow us to do the following three things: add rows together, multiply an entire row by a number, and swap the location of two rows. Often we will combine the first two operations into a single action. Through the use of row operations, we turn the left half of the augmented matrix into an identity matrix. Gauss-Jordan elimination tells us to begin by making the main diagonal nothing but 1's, and everything below the diagonal 0's.




    1
    1
    −2
    3
    2
    3
    −1
    2
    −1
    2
    4
    −15




    −2R1+R2
    R1+R3




    1
    1
    −2
    3
    0
    1
    3
    −4
    0
    3
    2
    −12




    −3R2+R3




    1
    1
    −2
    3
    0
    1
    3
    −4
    0
    0
    −7
    0




    1

    7
    R3




    1
    1
    −2
    3
    0
    1
    3
    −4
    0
    0
    1
    0




  • Now that the main diagonal is 1's and there are 0's below, we reverse and work back up the main diagonal, turning all the numbers above the main diagonal to 0's as well.




    1
    1
    −2
    3
    0
    1
    3
    −4
    0
    0
    1
    0




                            
    2R3+R1
    −3R3 + R2




    1
    1
    0
    3
    0
    1
    0
    −4
    0
    0
    1
    0




                            
    −R2+R1




    1
    0
    0
    7
    0
    1
    0
    −4
    0
    0
    1
    0




  • At this point the Gauss-Jordan elimination process has given us a matrix in reduced row-echelon form, and we can turn the augmented matrix back into a linear system:
    1a
    + 0b
    +0c
    =
    7
    0a
    + 1b
    +0c
    =
    −4
    0a
    + 0b
    +1c
    =
    0
a=7,   b=−4,   c=0
Using Gauss-Jordan elimination and a graphing calculator (or computer matrix calculator), solve the linear system below.
3p
−2r
+s
+2t
=
21
p
−2q
+3r
+t
=
18
+q
+6r
−3s
−2t
=
16
−3p
+3q
+4r
−2s
−t
=
3
p
+q
+r
+s
+t
=
14
  • It is possible to do this problem without using a graphing calculator or any sort of matrix calculating aid. That said, since it is such a large linear system, using one will make this problem much easier. If we are using such an aid, all we have to do is set it up properly and enter it in correctly. Begin by converting the linear system into an augmented matrix:







    3
    0
    −2
    1
    2
    21
    1
    −2
    3
    0
    1
    18
    0
    1
    6
    −3
    −2
    16
    −3
    3
    4
    −2
    −1
    3
    1
    1
    1
    1
    1
    14







  • The Gauss-Jordan elimination method allows us to turn the augmented matrix into its reduced row-echelon form (where the left side is an identity matrix). Most graphing calculators have some sort of function that allows you to convert it into that form. It is unlikely to be a button on the calculator, but is something you can find if you search the menus. [For example, on most Texas Instruments graphing calculators, you can find a menu that deals specifically with matrices, then choose the `rref' function and apply it to the matrix.] You may need to find specific instructions for how to use your calculator in such a way. You can probably find out how with a simple internet search of "[name/model of your graphing calculator] reduced row echelon matrix", or something similar. If you do not have a graphing calculator, you can also search the internet for "reduced row echelon matrix calculator"-there are plenty you can use with just a web browser.
  • Carefully enter the augmented matrix (it won't have a vertical bar before the sixth column) into your calculator and apply the reduced row-echelon function to it. Your calculator will churn out the following result:







    1
    0
    0
    0
    0
    5
    0
    1
    0
    0
    0
    2
    0
    0
    1
    0
    0
    3
    0
    0
    0
    1
    0
    −4
    0
    0
    0
    0
    1
    8







    We can then convert this into the value for each variable based on location.
p=5,   q=2,   r=3,   s=−4,   t=8
Using Cramer's Rule, solve the linear system below.
2x
−7y
=
−14
−5x
+6y
=
−11
  • We can solve a linear system through Cramer's Rule. First, start off by converting the coefficients of the linear system into a coefficient matrix. (Notice that this does not include the constants on the right side of the equations.)
    A =


    2
    −7
    −5
    6



  • Depending on which variable we're interested in solving for, we swap the column of constants for the corresponding column of coefficients from that variable:
    Ax =


    14
    −7
    11
    6



          
          Ay =


    2
    14
    −5
    11



    To actually find the solution for a variable, Cramer's Rule says it is given by the determinant of the modified matrix for that variable divided by the determinant of the non-modified coefficient matrix:
    x =
    det
    Ax

    det
    A
              
               y =
    det
    Ay

    det
    A
  • Thus, to find the solutions, we need to calculate detA, detAx, and detAy. [If you are unfamiliar with calculating determinants, make sure to check out the previous lesson.]
    det
    A     =    


    2
    −7
    −5
    6



        =    2 ·6 − (−7)·(−5)    =    −23

    det
    Ax     =    


    −14
    −7
    −11
    6



        =    −14·6 − (−7)·(−11)    =    −161

    det
    Ay     =    


    2
    −14
    −5
    −11



        =    2 ·(−11) − (−14)·(−5)    =    −92
    Thus, by Cramer's Rule, we have
    x =
    det
    Ax

    det
    A
      =  −161

    −23
      =  7          
               y =
    det
    Ay

    det
    A
      =  −92

    −23
      =  4
x=7,   y=4
Using Cramer's Rule, find the value of d in the below linear system. You may (and are strongly encouraged) to use a graphing calculator or computer matrix calculator to find the value of the determinants involved.
−4a
+4b
+2c
−2d
+5e
=
−6
3a
−b
−2c
−5d
+5e
=
37
−4a
−3b
−2c
+5d
+2e
=
16
−a
−4b
+5c
−3d
−e
=
−31
3a
−2b
−4c
−d
+e
=
32
  • We can solve a linear system through Cramer's Rule. First, start off by converting the coefficients of the linear system into a coefficient matrix. (Notice that this does not include the constants on the right side of the equations.)
    A =






    −4
    4
    2
    −2
    5
    3
    −1
    −2
    −5
    5
    −4
    −3
    −2
    5
    2
    −1
    −4
    5
    −3
    −1
    3
    −2
    −4
    −1
    1







  • Depending on which variable we're interested in solving for, we swap the column of constants for the corresponding column of coefficients from that variable. In this case, we're interested in solving for d, so we swap in the column of constants for the column of coefficients that match up to d.
    Ad =






    −4
    4
    2
    6
    5
    3
    −1
    −2
    37
    5
    −4
    −3
    −2
    16
    2
    −1
    −4
    5
    31
    −1
    3
    −2
    −4
    32
    1







    To actually find the solution for a variable, Cramer's Rule says it is given by the determinant of the modified matrix for that variable divided by the determinant of the non-modified coefficient matrix:
    d =
    det
    Ad

    det
    A
  • Thus, to find the solutions, we need to calculate detA and  detAd. Notice that it would take a lot of effort to find these determinants by hand since they are coming from 5 ×5 matrices. Luckily, the problem told us that we could (and should) use a graphing calculator or matrix calculator. If we've got some sort of computing device to do this process for us, it's a breeze. If you're using a graphing calculator, there will be some sort of det function in a menu about matrices. Choose that, then apply it to the matrix (which you will also have to enter). If you're using a matrix calculator on your computer (just do an internet search for "matrix calculator"), the process will likely be reversed: enter the matrix then tell it to take the determinant. However you're doing it, the results should be as follows:
    det
    (A) = 2067          
               det
    (Ad) = 6201
    Therefore, by Cramer's Rule, we have
    d =
    det
    Ad

    det
    A
        =     6201

    2067
        =     3
d=3
Using an inverse matrix, solve the system of linear equations below.
3x
−3y
=
9
−3x
+5y
=
−7
  • Notice that, by the rules of matrix multiplication, we can rewrite the above linear system as the interaction of matrices:
    3x
    −3y
    =
    9
    −3x
    +5y
    =
    −7
           ⇔       


    3
    −3
    −3
    5






    x
    y



    =


    9
    −7



    Symbolically, we can express this as
    AX = B,
    where A is the coefficient matrix from the system, X is a column matrix of the variables, and B is a column matrix of the constants on the right side of the equations. Therefore, if A has an inverse matrix A−1, we can use algebra to apply that inverse to the equation and cancel out the A, leaving X isolated on one side. Thus, if we multiply by A−1 on both sides, we get
    A−1   AX     =     A−1   B        ⇒        X = A−1 B
    Thus, if we find A−1, we can easily solve for X, which will tell us the solutions to the linear system.
  • From the previous lesson, we saw that it was quite easy to find the inverse of a 2×2 matrix. For any matrix A, we have
    A =


    a
    b
    c
    d



        ⇒       A−1 = 1

    ad−bc



    d
    −b
    −c
    a



    Thus, we can find the inverse to the coefficient matrix by just plugging in:
    A−1 = 1

    3·5−(−3)(−3)



    5
    3
    3
    3



        =    





    5

    6
    1

    2
    1

    2
    1

    2






  • Finally, by the logic we talked about in the first step, we have
    X = A−1 B     =    





    5

    6
    1

    2
    1

    2
    1

    2









    9
    −7



        =    





    15

    2
    7

    2
    9

    2
    7

    2






        =    


    4
    1



    Therefore, since X represented a column matrix of the variables, we have shown



    x
    y



    =


    4
    1



x=4,   y=1
Using the ideas behind solving a linear system with an inverse matrix, show that the below linear system does not have a unique solution (that is, it has either infinitely many or no solutions).
2x
−2y
−3z
=
−2
−3x
+3y
+5z
=
4
2x
−2y
−5z
=
−14
  • As in the previous problem, we can use matrix multiplication to rewrite the system:
    2x
    −2y
    −3z
    =
    −2
    −3x
    +3y
    +5z
    =
    4
    2x
    −2y
    −5z
    =
    −14
           ⇔       



    2
    −2
    −3
    −3
    3
    5
    2
    −2
    −5








    x
    y
    z




    =



    −2
    4
    −14




    Symbolically, we can express this as
    AX = B,
    where A is the coefficient matrix from the system, X is a column matrix of the variables, and B is a column matrix of the constants on the right side of the equations. Therefore, if A has an inverse matrix A−1, we can use algebra to solve for X
    A−1   AX     =     A−1   B        ⇒        X = A−1 B
    From the above, we see that if we can find A−1, we can easily find a unique solution for X, which means we have found a unique solution for the linear system.
  • However, if we can not find an inverse for A, then it is not possible for us to find a unique solution to the linear system. Therefore, to show that the linear system does not have a unique solution (which is what the problem told us to do), we only need to show that A is not invertible. From the previous lesson on determinants and inverses, we know that a matrix is not invertible when its determinant equals 0. Thus, we need to show that det(A) = 0.
  • Take the determinant of A [If you're not sure how the below works, make sure to check out the previous lesson on determinants.]:
    det
    (A)     =    



    2
    2
    3
    −3
    3
    5
    2
    −2
    −5





    2 ·


    3
    5
    −2
    −5



        −(−2) ·


    −3
    5
    2
    −5



        +(−3) ·


    −3
    3
    2
    −2




    2·(3·(−5) − 5·(−2))   +  2 ·((−3)·(−5) − 5·2)   +  (−3) ·((−3)·(−2) − 3 ·2)

    2·(−5) + 2 ·(5) −3 ·(0)     =     0
    Thus det(A) = 0, so we have shown that the linear system does not have a unique solution because we cannot invert A.
Show that the coefficient matrix for the linear system has a determinant of 0. [Notice that this does not prove that the linear system has no solutions at all, it only shows that it has no unique solutions. It could have infinitely many solutions or no solutions whatsoever. All we know from det(A) = 0 is that the system does not have a unique solution.]
Using an inverse matrix, solve the system of linear equations below. [Use a graphing calculator or matrix calculator to find the inverse matrix.]
2a
+3b
+5c
+4d
−2e
=
48
a
+2c
−3d
+e
=
−14
−a
+2b
+3c
−2d
+e
=
−11
3b
−3c
+3d
−3e
=
18
a
+2b
+4c
−7d
+e
=
−34
  • Using matrix multiplication, we can rewrite the linear system:







    2
    3
    5
    4
    −2
    1
    0
    2
    −3
    1
    −1
    2
    3
    −2
    1
    0
    3
    −3
    3
    −3
    1
    2
    4
    −7
    1














    a
    b
    c
    d
    e







    =






    48
    −14
    −11
    18
    −34







    Symbolically, we can express this as
    AX = B,
    where A is the coefficient matrix from the system, X is a column matrix of the variables, and B is a column matrix of the constants on the right side of the equations. Therefore, if A has an inverse matrix A−1, we can use algebra to solve for X
    A−1   AX     =     A−1   B        ⇒        X = A−1 B
  • Thus, to solve the linear system, we only need to find A−1 B using a graphing calculator or matrix calculator. One possibility would be to find A−1 using the calculator, then enter that result in and multiply it by B. However, that will require a lot of work entering two different 5 ×5 matrices (one for A, then again for A−1 once we know it). Instead, we can usually speed up the process by telling the calculator to do both steps one after the other. Enter in to the calculator something along the lines of







    2
    3
    5
    4
    −2
    1
    0
    2
    −3
    1
    −1
    2
    3
    −2
    1
    0
    3
    −3
    3
    −3
    1
    2
    4
    −7
    1







    −1








     
      






    48
    −14
    −11
    18
    −34







    Doing this will cause the calculator to both find the inverse and then immediately multiply that by the constant matrix B. [However, depending on the specific calculator you are using, you might have to enter in A−1 by hand. But it is likely you can find some method to avoid that extra work.]
  • However you wind up doing it, the calculator will give that







    2
    3
    5
    4
    −2
    1
    0
    2
    −3
    1
    −1
    2
    3
    −2
    1
    0
    3
    −3
    3
    −3
    1
    2
    4
    −7
    1







    −1








     
      






    48
    −14
    −11
    18
    −34







        =    






    2
    −1
    3
    6
    −4







    Thus, since X = A−1 B, we have that







    a
    b
    c
    d
    e







        =    






    2
    −1
    3
    6
    −4







a=2,   b=−1,   c=3,   d=6,   e=−4

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Using Matrices to Solve Systems of Linear Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:12
  • Augmented Matrix 1:44
    • We Can Represent the Entire Linear System With an Augmented Matrix
  • Row Operations 3:22
    • Interchange the Locations of Two Rows
    • Multiply (or Divide) a Row by a Nonzero Number
    • Add (or Subtract) a Multiple of One Row to Another
  • Row Operations - Keep Notes! 5:50
    • Suggested Symbols
  • Gauss-Jordan Elimination - Idea 8:04
    • Gauss-Jordan Elimination - Idea, cont.
    • Reduced Row-Echelon Form
  • Gauss-Jordan Elimination - Method 11:36
    • Begin by Writing the System As An Augmented Matrix
    • Gauss-Jordan Elimination - Method, cont.
  • Cramer's Rule - 2 x 2 Matrices 17:08
  • Cramer's Rule - n x n Matrices 19:24
  • Solving with Inverse Matrices 21:10
    • Solving Inverse Matrices, cont.
  • The Mighty (Graphing) Calculator 26:38
  • Example 1 29:56
  • Example 2 33:56
  • Example 3 37:00
    • Example 3, cont.
  • Example 4 51:28

Transcription: Using Matrices to Solve Systems of Linear Equations

Hi--welcome back to Educator.com.0000

Today, we are finally going to see why we have been studying matrices--just how powerful they are.0002

We are going to use matrices to solve systems of linear equations.0006

Consider the following system of linear equations: x + y + z = 3; 2x - 2y + 3z = -4; and -x - z = 0.0010

Notice that, as long as we keep the variables in the same order for each equation (we can't swap it0019

to y + x + z; we keep it in xyz order every time), we could write these coefficients to all of the variables as a coefficient matrix.0023

What we have in front of this x is just a 1; in front of this y is just a 1; and in front of this z is just a 1.0031

So, we can write a first row of 1, 1, 1, all of these coefficients that are on that row right there of the equation.0037

For the next one, we have 2, -2, 3; so we put them all down here; so we have done that equation as the coefficients in that row, showing up there.0045

And since we can always trust that we are going to have the x, the y, and the z here,0055

because we are always staying in this order of x, y, z (that is why we have to keep the variables in the same order each time),0060

we can create this coefficient matrix; and finally, we have -1 here and -1 here.0068

And why do we have a 0? Well, if y doesn't show up, it must be because we have a 0y; so that is why we get a 0.0074

So, we have done all 3 equations, the coefficients to the variables in all three equations.0081

This idea of converting the information in a linear system into a matrix0085

will allow us to explore ways that we can have linear systems interact with matrices and vice versa.0089

How can a matrix allow us to solve a linear system of equations?0094

Our first idea is the augmented matrix; we can take this idea of the coefficient matrix and expand on it.0098

Instead of just representing coefficients for the equations, we can represent an entire linear system,0104

the solutions included, with an augmented matrix.0109

So, previously we didn't have the constants for the equations, what was on the right side of the equals sign.0112

So now, we have that show up on the right side over here.0118

So now, we have the coefficients, and we have what each of those equations is equal to.0121

So, each row represents an equation of the system: x + y + z = 3 is 1, 1, 1, 3, because that first 10126

represents the coefficient on x; the first one on the y; the first one on the z; and it all comes together to equal 3,0136

because we know that just addition is what is going on between all of those coefficients, because it is a linear system.0142

Each left-side column gives a variable's coefficients: all of the coefficients to x show up here.0147

We have 1, 2, and -1 on the x, and we have 1, 2, and -1 in that column, as well, on the left side.0153

Where the variable does not appear, it has a coefficient of 0.0160

The 0 is to show that we have nothing for y here, because if you have nothing, we can think of it as just 0 times y.0163

The vertical line represents equalities; since this vertical line here is representing each of these, there is an equals sign here.0171

And finally, the right-side column, this column here, gives us...0179

This column of our matrix is the constant terms from our equations.0183

So, we have a way of converting that entire set of equations into a single matrix.0187

So now, we can look at how we can play around with this matrix to have it give up what those variables are equal to.0192

Since we can represent a linear system as an augmented matrix, we can do operations on the matrix the same way we interact with a linear system.0199

Anything that would make sense to do to a linear system, to an equation in a linear system,0206

should make sense--there should be a way to do it over on the matrix version,0209

because since we can do it to a linear system, and our augmented matrix is just showing us a linear system--0213

it is a way to portray a linear system--then if we can figure out a way to interact with our matrix0220

that is the same as interacting with a linear system, we know that it is just fine.0224

So, that gives us the idea of 3 row operations.0228

The first one is to interchange the locations of two rows.0231

If I have row 1 and row 2, I can swap their places.0235

Our next idea is to multiply or divide a row by a non-zero number.0239

I can multiply an entire row by 2 or by 5 or by -10--whatever I want to--0243

or divide by 2, because that is just the same as multiplying by 1/2.0249

And then finally, add or subtract a multiple of one row to another.0253

If I have row 2 here and row 5 here, I can have row 5 subtract twice on itself; so it is row 2 - 2(row 5).0257

I can have a multiple of one row subtract from another one.0267

So, why does this make sense--how is this like a linear system?0270

Well, all of these operations are completely reasonable, if we had a linear system.0274

If we have an equation here and an equation here, it is totally meaningless for us to swap the order that the equations come in.0278

The location of the equation isn't an important thing when we are working with a linear system.0284

We just have to look at all of them, so it doesn't matter which came first and which came last.0289

So, we can move them around, and it doesn't matter.0292

That means that we can move our rows around, and it doesn't matter, because it is just representing a linear system.0294

Similarly, multiplying both sides of an equation is just algebra.0298

If I have an equation, I can multiply 2 on the left side and 2 on the right side, and that is just fine.0302

You can think of that as multiplying each of the numbers inside of the equation.0308

If we multiply each of the numbers inside of the equation, that is the same thing as just multiplying an entire row of our augmented matrix.0313

Finally, adding a multiple of an equation is elimination.0318

Remember: when we were talking about linear systems at first, elimination was when we can add a multiple of one equation to another equation.0322

Well, that is the same thing as adding a multiple of one row to another row over in the augmented matrix version.0327

So, everything here has a perfect parallel between the two ideas.0333

Each one of our row operations makes total sense, if we were just working with a linear system.0336

And since our augmented matrix is just representing a linear system, they make sense over here with the augmented matrices, as well.0341

While row operations are a simple idea (each one of these is pretty simple--swap; multiply; or add a multiple--not that crazy)--0348

working with them will involve a lot of arithmetic and steps.0356

You are going to have a lot of calculations going on, and while none of them will probably be very hard calculations,0360

you are going to be doing so many that it is easy to make mistakes.0364

So, when you are working on row operations, when you are working on doing this stuff,0368

I want you to be careful with what you are doing, and counter the fact that you are likely to make mistakes0372

by noting each step: note what you just did for each step.0376

In addition to this being a good way to keep you from making mistakes, some teachers will simply require it,0380

and won't give you credit if you don't do it; and that is pretty reasonable,0384

because you will definitely end up making mistakes sooner or later if you don't do this sort of thing.0386

So, take a note of what you did on each step; show what you did between the first one, and then the second one,0391

and then the third one, by writing on the side what you just did.0397

This will make it easier to avoid making mistakes, and it will help you find any that manage to creep through.0401

If you get to the end, and you see that this doesn't make sense--something must have gone wrong--0405

you can go up and carefully analyze each of your steps, and figure out where you made a mistake.0408

Or maybe things actually do come out to be weird for some reason.0413

So, here are some suggested symbols for each of the three row operations.0416

You don't have to use them; use whatever makes sense to you (and if your teacher cares about it, whatever makes sense to your teacher).0420

But these work well for me, and I think they make sense, pretty clearly.0424

If we are interchanging row i and row j (we are swapping their locations), I like just a little arrow, left-right, between them.0428

We show some row by using a capital R to talk about a row, and then the number of it.0435

For example, if we want to talk about the second row, we would just talk about it as R2.0440

If we wanted to talk about the ninth row, it would be R9; and so on, and so on.0444

So, if we are swapping row i and row j, we have Ri, little arrow going back and forth, Rj.0450

If we want to multiply row i by the number k (we are multiplying by k), we just have k times row i--as simple as that.0455

And finally, if we are adding a multiple of row i to row j, then we have kRi (k times row i,0463

the thing that we are adding the multiple of) to what it is being added to.0472

We will see these pretty soon, when we are actually starting to see how this stuff gets done.0477

Gauss-Jordan elimination: here is an idea: since all of our row operations make sense for solving a linear system--0482

they all make sense for a way to do it--we can apply them to find the value of each variable.0489

If we manage to get our augmented matrix in a form like this form right here, we would immediately know what each variable is; why?0494

Well, notice: this first row here has 1, 0, 0; well, 1 here would correspond to the x here.0505

And then, this one here would be non-existent, because there are two 0's there; so there would be no y; there would be no z.0511

And we know that it is equal to -17.0516

The exact same thing is going on here; we know that y, because that is the y column, must be equal to 8 (nothing else shows up),0520

and that z (since that is the z column) must be equal to 47.0528

We have managed to solve the system by just moving stuff around in this augmented matrix.0531

We know that that has to be true, because that augmented matrix must be equivalent to the linear system,0536

because we turned our linear system into an augmented matrix, and then we had all of these row operations0541

that are just the same as working with a linear system.0546

So, what we have here is still the same as our original linear system.0548

So, we can convert back to the linear system and see what our answers are.0551

We call this format for a matrix reduced row echelon form; it is a mouthful, and it is kind of confusing at first.0556

Echelon has something to do with a triangle shape; it is "row echelon" because the rows are kind of arranged in a triangle,0563

and "reduced" because they all start with 1's, and there are 0's...0569

Honestly, don't really worry about it; just know reduced row echelon form, that really long name, and you will be fine.0572

While it has a formal definition (there is a way to formally define it), it is going to be enough0577

for us to just think of it in this casual way, where it is a diagonal of 1's.0581

That main diagonal will have all 1's on it; it will start at the top left, and it will continue down diagonally.0585

And it will have 0's above and below.0591

So, we have 1's along the main diagonal, and above and below our 1's, there will be 0's.0593

So, we have a 0 here above the 1's and a 0 here below the 1's.0603

We have 0's here above the 1's and a 0 here below the 1's.0607

Finally, the numbers on the far right can be any number.0611

Over here, -5, 8...it doesn't matter; over here, we manage to have two columns,0616

because the 1 diagonal's part of the reduced row echelon form is just an identity matrix.0620

So, it can't get any farther than whatever the square portion of it would be.0628

So, we can end up having multiple columns, as well.0632

But in our case, when we are working on this elimination to solve linear systems, we will always only have a single column on the far right.0634

Anyway, the point is that we have this diagonal of 1's with 0's above and below; and the stuff on the far right can be any number.0641

Now, from what we have just discussed, if we can use row operations to put an augmented matrix in reduced row echelon form,0648

like this one right here, we will solve its associated linear system,0653

because we will know that, since there is only one of a variable in that row, it must be equal to the constant0657

on the other side of the augmented matrix--one the right side of the augmented matrix, past that vertical line that shows equality.0662

We will have solved its associated linear system.0669

Gauss-Jordan elimination is just named after the people who created it.0672

It is a method we can follow to produce reduced row echelon form matrices through row operations to solve linear systems.0676

It is just a simple method of being able to get 0's to show up and 1's to show up on the diagonal, and then we are done.0683

So, it is just a method that we can follow through that will always end up resulting in a reduced row echelon form.0689

All right, so let's see how it is done.0695

The very first thing you do for Gauss-Jordan elimination is: you take your linear system, and you write it in augmented matrix form.0697

So, we have our augmented matrix form over here.0705

We look at the coefficients; we convert them over; the line to show the equals signs is here, and then our constants are on the far right side.0707

All right, the next thing: you use row operations to attain a 1 in the top left0715

(we start in the top left here), and then 0's below.0722

We get a 1 here; and then we wanted 0's below it, because it is a 1, and 0's are above and below.0726

So, we start working with 1's, creating 1's...well, sorry, from your point of view...creating 1's and creating 0's underneath them.0731

So, we first get a 1 up here; and then we create the 0's underneath it.0740

Once we have done that, we move on to creating the next one diagonally down and doing 0's below that.0746

And we just keep repeating until we have made it all the way down the diagonal.0751

So, you just keep going until you are all the way down the diagonal, creating 1's and creating 0's below.0754

First, we started with a 1 up here; we already have the first thing done.0760

So, our next step is...how do we get this stuff to turn into 0's?0764

We do row operations: we want to get rid of this 2, so since row 1 has a 1 there, we subtract by 2(row 1).0767

2 row 1 gets us -2 here, -2 here; this becomes a 0; -2 times 1 on -2 gets us -4; 1 times -2, added to 3, gets us 1; and -2 times 3, added to -4, gets us -10.0776

Next, we are adding row 1, because we need to get rid of this -1 here.0795

So, we add row 1, because it is positive 1; add +1 to -1; we get 0; add +1 to 0; we get 1; add +1 to -1; we get 0; add +3,0800

because it is just a multiple--how many times we are adding whatever is on that first row, because it was just 1 of row 1-- 3 on 0 gets us 3.0813

All right, at this point, we have 0's below; great.0821

So now, we are ready to move on to the next step in the diagonal.0824

All right, our next step in the diagonal is going to be this -4 here.0828

We want to get that to turn into a 1.0832

We could do this by manipulating it with canceling things out, or by dividing both sides of that entire row by -4,0834

or multiplying the entire row by -1/4; but we might notice that we already have a 1 here.0844

Well, let's just use that: if we swap these two rows, we will manage to have a 1 in our next location; great--we do that instead.0850

So, we make row 2 swap with row 3, because up here, they used to be row 2 and row 3.0858

And now, they take their new spots; they swap locations; and we have our new matrix right here.0864

The next step: we see that we have the 1 here, so our next step is...we need to turn everything below the 1's to 0's on this first portion.0871

So, how do we do that? Well, we can add +4 times row 2, because we have a 1 here.0882

So, we add 4 times that, and that will cancel out the -4.0887

4 times 1, added to this, gets us 0; 0 times 4, added to 0--we still have 0.0891

0 times 4, added to this--we just have 1 still; 3 times 4, added to -10, gets us 2.0897

So, at this point, notice that we have nothing but 1's on our main diagonal, and we have 0's all below it.0903

However, we don't have 0's above it yet; there are things other than that; so that is the next step.0910

Once you have 1's all along the main diagonal and 0's all along below that main diagonal,0917

the next step is to cancel out the stuff above it.0927

Our first thing is: we work from the bottom right of the diagonal, and we cancel out above the 1's.0930

You work your way up: so our first step is to cancel out everything here and here.0940

But we notice that, because this row is 0, 1, 0, we can actually do it in one step,0944

where we can subtract one of row 3 and one of row 2 (it is getting kind of hard to see, with all of those colors there).0948

So, subtracting one of row 3: we have 1 here, minus 1; so now we subtract 2 on this, so we have 3 - 2 so far,0954

for what is going to show up here; let's also subtract by row 2; row 2...minus here...0964

1 - 1 comes out to be 0; since it is zeroes everywhere else on that row, we don't have to worry about them interfering,0971

except for over here; we have it subtracting by another 3; so 3 - 2 - 3 comes out to be equal to -2, and we get -2 here.0976

At this point, we have reduced row echelon form.0986

We have 1's on the main diagonal and 0's above and below, so we can convert this.0990

Our x here becomes -2; the representative -1 of y here becomes 3, so we have y = 3.0995

And the representative 1z becomes z = 2; so now we have solved the thing.1002

And I want you to know: if there is no solution, or infinitely many--if our linear system can't be solved,1007

or it has infinitely many solutions--this method will end up not working.1013

You will not be able to achieve reduced row echelon form if there is no solution or infinitely many solutions.1017

So, that is something to keep in mind.1023

All right, a new way to do this: we are going to look at a total of 3 different ways to solve linear systems, each using matrices.1026

Another way to do this is through Cramer's Rule.1033

We can find the solutions to a linear system by this rule.1035

Given a two-variable system (we will start with 2 x 2, until we get a good understanding of what is going on),1037

where we have the a's (each of our a's here is just a constant), a11x, a12y, a21x, a22y,1042

is equal to constants on the right side (b1 and b2 are also constants),1050

we can create a normal coefficient matrix (A is the normal coefficient matrix).1055

All of our coefficients, a11, a12, a21, a22...1063

show up just like they would normally in a coefficient matrix.1068

Now, Ax...what it is going to do is take this column here on a, and it is going to replace it with the constants to the equation.1071

It is going to replace a1, a21 with this; and so, we have b1, b2 for that column.1083

And then, the rest of A is like normal.1089

Similarly, for y, we are going to swap out the y constants from A.1092

And so, we are going to have A like normal, except we swap out the constant column for where the y variables were occupying.1098

The y column gets swapped to the constant column.1105

So, the constant column goes in there.1108

Notice that Ax is just like A, except that it has this constant column replacing the x column.1111

Similarly, Ay has the constant column replacing the y column from our normal coefficient matrix.1119

All right, that is the idea; now if the system has a single solution, if it comes out to be just one solution--1125

it isn't infinitely many; it isn't no solutions at all; if the system has a single solution,1131

then x will be equal to the determinant of Ax, over the determinant of A,1137

the determinant of its special matrix, divided by the determinant of the general coefficient matrix.1143

Similarly, y is going to be equal to the determinant of its special matrix, divided by the determinant of the general coefficient matrix.1150

OK, this method can be generalized to any linear system with n variables.1160

Let A be the coefficient matrix for this n-variable linear system.1165

Let Ai be the same as A, except the ith column; the column that represents1169

the variable we are currently working with--the variable that we want to solve for--that column1177

will be replaced with the column of constants.1182

So, we replace the column for the variable we are interested in solving for with the column of constants.1185

And that makes A for whatever variable we were looking for: Ai in this case,1191

if we are looking for the ith variable, up until we are looking for the ith one.1195

We replace it with the column of constants from the linear system, just the right side of the equations,1202

the b1, b2 on our previous 2 x 2 example.1206

OK, with that idea in mind, if the determinant of A, the determinant of our normal coefficient matrix, right up here,1210

is not equal to 0, then the ith variable, xi, this variable we are trying to solve for,1216

is equal to the determinant of its special matrix that has that column replacing it,1223

divided by the determinant of the normal coefficient matrix; that is what we have right here.1227

It is just like in the 2 x 2 form, except we can do it on a larger scale, as well.1232

You swap out this one column; you take the determinant of that special matrix; you divide it by the determinant of the normal coefficient matrix.1236

And that gives you the variable for whatever column you had swapped.1242

We will get the chance to see this done on a more confusing scale (which is the sort of thing1247

that we want to be able to understand--this on a larger scale) in Example 3.1251

And we will just see this get applied normally in Example 2 for a 2 x 2 matrix.1255

Also notice: if the determinant of A is equal to 0, then the system will have either no solutions or infinitely many solutions.1259

All right, the final method to do this: we can solve with inverse matrices.1266

This one is my personal favorite for understanding how this stuff works; I think it is the easiest to understand.1271

But that is maybe just me.1276

Using matrix multiplication, we can write a linear system as an equation with matrices.1278

How can we do this as an equation?1282

It made sense with an augmented matrix, because we talked about the special thing.1284

But how is 1, 1, 1, 2, -2, 3, -1, 0, -1--notice that that is just our normal coefficient matrix A showing up here--1287

if we multiply it by the column matrix x, y, z equals our coefficient column here, that ends up being just the same--1298

it is completely equivalent; these two ideas here are completely equivalent--multiplying the matrices1313

versus the linear system, and the linear system versus the matrices being multiplied together; they are completely the same.1319

Let's see why; let's just do some basic matrix multiplication on this.1323

What is going to come out of this? We have a 3 x 3 (3 rows by 3 columns), 3 rows by 1 column;1327

so yes, they match up, so they can multiply; that is going to produce a 3 x 1, 3 row by 1 column, matrix in the end.1335

So, let's see what is going to get made out of this.1342

We are going to have a 3 x 3; our first row times the only column is 1, 1, 1, 1; so I'll make this a little bit larger,1346

so we can see the full size of what is going to go in...1 times x + 1 times y + 1 times z is x + y + z.1356

Next, 2, -2, 3 on x, y, z gets us 2x - 2y + 3z.1369

Finally, -1, 0, -1 on x, y, z gets us -x + 0y (so let's just leave it blank) - z.1378

Now, if we know that that is equal to our coefficient matrix, because we said it from the beginning,1389

then all we are saying...3, -4, 0...well, for two matrices to be the same thing, for them to be equal to each other,1395

every entry in the two matrices has to be equal to its entry in the same location.1403

So, the top one, x + y + z, equals 3; that is just the exact same thing as this.1409

2x - 2y + 3z has to be equal to -4; well, that is the same thing as saying 2x - 2y + 3z = -4.1416

And the same thing: -x - z is saying it is equal to 0 through the matrices; and that is the same thing it was saying by the linear system.1424

So, the linear system, taken as a whole, is just the same thing as taking the coefficient matrix,1430

multiplying it by this coefficient column matrix...and that is going to come out to be equal to our constant matrix,1434

which was the constants for the equations; that is the idea that is going to really be the driving force behind using inverse matrices.1444

All right, we can symbolically write this whole thing as Ax = B; A times x equals B,1451

where A is the coefficient matrix right here; then X is a single-column matrix of the variables;1463

that is our x, y, z; whatever variables we end up using are going to go like that; and then finally,1476

B is a single-column matrix of the constants (3, -4, 0 gets the same thing right here); OK.1484

Notice: if we could somehow get X alone, if we could get our variable matrix, our variable column, alone on one side,1492

whatever it was on the other side, if it equals numbers on the other side in a matrix,1499

then we would have solved for it, because we would say that x is equal to whatever the corresponding location is on the other side;1503

y is equal to whatever its corresponding location is on the other side; z is equal to whatever its corresponding location is on the other side.1508

We would have solved for this.1514

So, if we can somehow get X alone, we will be done; we will have figured out what x, y, and z are equal to.1515

How can we do that, though--how can we get rid of A? Through inverse matrices!1521

That is no surprise, since this thing is titled Inverse Matrices.1525

We cancel out A; if A is invertible, then there exists some A-1 that we can multiply that by that will cancel out A.1527

So, we started with Ax = B; we can multiply by A-1 on the left side on both sides.1537

Remember: if you multiply by the left and the right, for matrix equations that doesn't work.1542

You have to always multiply both from the left or both from the right.1548

You are not allowed to do them on opposite sides; they have to both be coming from the same side when you multiply.1553

We multiply by A-1 on the left side on both cases; the A-1 here and the A cancel out, and we are left with just X = A-1B.1558

So, if we can compute A-1, and then we can compute what is A-1 times B, we will have solved our system.1566

We will have what our system is equal to.1573

Just make sure that you multiply from the same side for your inverse on both sides of the equation.1575

You have to multiply both from the left; otherwise it won't work out.1579

Finally, if A is not invertible (if the determinant of A is equal to 0, then you can't invert it),1582

then that means that the system has either no solutions or infinitely many solutions.1588

All right, let's try putting these things to use.1593

Oh, sorry; before we get into using them, the mighty graphing calculator:1596

all of these methods work great for solving linear systems: augmented matrices with Gauss-Jordan elimination,1601

Cramer's Rule, inverse matrices--they are all great ways to solve linear systems.1606

But they all have the downside of being really tedious; they take so much arithmetic to use.1611

We can work through it; we can see that we can do this stuff; but it is going to take us forever to actually work through this stuff by hand.1617

I have great news; it turns out that, if you have a graphing calculator, you can already do this right now, really fast, really quickly, and really easily.1623

Almost all graphing calculators have the ability to do matrix and vector operations.1631

You can enter matrices into your calculator, and then you can multiply them; you can take determinants of the matrices;1636

you can find inverses; or you can put them in reduced row echelon form.1641

Look on your graphing calculator, if you have a graphing calculator, for something that talks about where...1645

just look for a button about matrices; look for something like that.1649

And it will probably have more information about how to create a matrix, and then how to do things with it.1652

Inverse is probably just raising the matrix to the -1, and it will give out the value.1657

Each graphing calculator will end up being a little bit different for how it handles inputting the matrices.1661

But they will almost all have this ability, for sure.1665

If you have real difficulty figuring out how to do it on your calculator, just do a quick Internet search for "[name of your calculator] put in matrices."1669

Use "matrices," and you will be able to figure out an easy way to do it very quickly.1675

Someone has a guide up somewhere.1678

Also, if you don't have a graphing calculator, don't despair; it is still possible to get this stuff done really easily and really quickly.1680

There are a lot of websites out there where you can do these things for free.1687

Just try doing a quick Internet search for matrix calculator; just simply search the words matrix and calculator,1690

and the first 5 hits or so will all be matrix calculators, where you can plug in matrices,1698

and you can normally multiply them, or you can take their determinants, or you can get their inverses.1704

Or you can do other things that you don't even know you can do with matrices yet.1708

But just look for the things that you are looking for; there are lots of things you can do with it.1711

Do a quick Internet search for the words matrix and calculator, and you will be able to find all sorts of stuff for those.1714

So, even if you don't have a graphing calculator, there are lots of things out there.1721

If you are watching this video right now, you can go and find websites that will let you do this for free.1724

Finally, while this is great that we can do all of this stuff with a calculator,1729

and the calculator will do the work for us, I still want to point out that it is important to be able to do this stuff without a calculator.1732

So, it makes it so much easier to be able to use a calculator; but we still have to understand what is going on underneath the hood.1738

We don't have to constantly be using it, but we have to have some sense of what is going on under the hood1744

if we are going to be able to understand more, higher, complex-level stuff in later classes.1749

So, you want to be able to understand this stuff, just because you want to be able to understand things,1754

if you are going to be able to make sense of things that come later.1757

And also, you usually need to show your work on your tests.1759

Your teacher is not going to be very happy if you are taking a test, and you just say, "My calculator said it!"1762

You are not going to get any points for that.1767

So, you can't just get away with it all the time.1769

That said, it can be a great help for checking your work, so you can work through the thing by hand,1771

and then just do a quick check on your calculator to tell you that you got the problem right; that is really useful on tests.1775

Or if you are dealing with really huge matrices, where it is 4 x 4, 5 x 5, 6 x 6, or even larger,1780

where you can't reasonably be able to do that by hand, you just use a calculator, and that is perfectly fine.1786

All right, now let's go on to the examples.1791

The first example: Using Gauss-Jordan elimination, solve 2x + 5y = -3, 4x + 7y = 3.1793

Our very first thing to do is: we need to convert it into an augmented matrix.1799

We have 2 as the first coefficient on the x, and then 5 as the first coefficient on the y, and that equals -3.1803

So, there is our bar there; 4, 7, 3; we have converted it into an augmented matrix.1809

Our coefficients are on the left part of the matrix, and our constant terms are on the right part of the matrix.1818

All right, at this point, we just start working through it.1825

The very first thing that we need to do is to get that to turn into a 1--get the top left corner to turn into a 1.1828

So, we will do that by multiplying the first row: 1/2 times row 1.1834

All right, that is what we will do there: 2 times 1/2 becomes 1; 5 times 1/2 becomes 5/2; -3 times 1/2 becomes -3/2.1839

4, 7, 3; the bottom row didn't get touched, so it just stays there.1849

The next thing to do: we want to get this to turn into a 0.1853

So, we will subtract the top row; the top row is not going to end up doing anything on this step,1858

but we will subtract the top row 4 times, because we have 4 here; so - 4R1 + our second row.1866

-4 times 1 plus 4 gets us 0; -4 times 5/2 gets us -10; -10 + 7 gets us -3; -3/2 times -4 gets us +6; we got +6 out of that, so that gets us +9.1877

Our next step: we want to get this to turn into a 1; we will bring this whole thing up here.1900

The next step is to get the second row to turn into a 1: 1, 5/2, -3/2.1908

We multiply the bottom part by -1/3 times row 2; so 0 times -1/3 is still 0; -3 times -1/3 becomes +1; 9 times -1/3 becomes -3.1916

At this point, we can now turn this into a 0; we don't need to do anything to our bottom row; it is still 0, 1, -3.1930

But we will add -5/2 of row 2 to row 1; so 0 times anything is still going to be 0; so added there...it is still 1 there.1940

Then, -5/2 on 1 + 5/2 becomes 0; -5/2 times -3 becomes +15/2, and then still -3/2.1953

Let's simplify that: we have 1, 0, 0, 1...15/2 - 3/2 becomes 12/2; 12/2 is 6...6, -3.1963

So, at this point, we can convert that into answers; x = 6; y = -3.1975

There are our answers; however, we did have to do a whole lot of calculation to get to this point.1983

And it could be even more if we were working on a larger augmented matrix; they get big really fast.1987

So, it might be a good idea to do a quick check; let's just check our work and make sure it is correct.1992

Let's plug it into the first one: 2 times 6, plus 5 times -3; what does that come out to be?1998

We hope it will come out to be -3: 12 + -15 = -3; indeed, that is true.2004

We could check it again with this equation, as well, if we want to be really, really extra careful.2011

4 times 6, plus 7 times -3, equals positive 3; 24 - 21 = 3; that is true.2015

So, both of our checks worked out; we know that x = 6, y = -3; that is definitely a solution--great.2025

All right, the second example: let's see Cramer's Rule in action.2031

The first thing we want to do, if we are going to use Cramer's Rule, is: we need to get a coefficient matrix going.2034

A =...what are our coefficients here? We have a 2 x 2: 2, 5, 4, 7.2039

There are the coefficients; our next step is...we want Ax--what is Ax going to be?2048

Here is our x column; we are going to swap that out for the constants here.2055

-3 and 3 replaces what had been our x column here.2062

And then, the rest of it is just like normal; so we replace that one column, but everything else is just the same.2067

Ay: what will Ay be?2073

The same sort of thing, except now we are replacing the y column--what is the y column going to turn into?2075

It is going to also become -3, 3; so 2, 4 is just as it was before; the first column is still the same, because that is the x column.2081

But now we are swapping out the y column, so it becomes the constants, -3 and 3.2088

All right, so we were told that Cramer's Rule says that x is equal to the determinant of its special, swapped-out matrix, Ax,2093

divided by the determinant of the normal coefficient matrix.2103

The determinant of -5, 3, 5, 7, divided by the determinant of 2, 5, 4, 7:2107

-3 times 7 gets us -21; minus 3 times 5 (is 15), over 2 times 7 (is 14), minus 2 times 5 (is 20);2118

we have -36/-6 =...that comes out to be positive 6, so we now have x = +6.2130

That checks out with what we just did in the previous one.2140

If you didn't notice, these equations are the same as what they were in the previous example,2142

so we are just seeing two different ways to do the same problem2146

(well, at least the part where we are trying to solve for x and y).2149

So, that checks out, because we checked it in the previous problem.2153

Now, what about y? y is going to be the same thing; y is equal to the determinant...same structure, at least...2155

of Ay, its special matrix, divided by the determinant of A, once again.2162

We could calculate the determinant of A, but we already calculated the determinant of A.2168

We figured out that it comes out to be -6, so we just drop in -6 here.2173

You only have to do it once; it is not going to change--the determinant of A will stay the determinant of A, as long as A doesn't change.2177

Then, the determinant of Ay: we don't know what Ay, that special matrix, is, yet.2183

2, -3, 4, 3: 2 times 3 is 6, minus 4 times -3 (-12); that cancels out; so we have 6 + 12,2187

divided by -6; 18/-6 equals -3; so y comes out to be -3.2197

Once again, that is the same answer as we had on the previous example, so we know that this checks out.2205

If you had just done this for the very first time, I would recommend doing a check,2210

because once again, you have to do a lot of arithmetic to get to this point.2213

And it is going to be even more if you are doing a larger Cramer's Rule, like, say, this one.2216

All right, so if we are working on this one, we only have to solve for the value of y.2220

There is one slight downside to only solving for one variable.2224

It means you can't check your work, because we can't plug in y and be sure that it works out to be true.2227

But we can at least get out what it should be.2231

All right, using Cramer's Rule, the first thing we need to do is figure out what our coefficient matrix A is.2234

A =...all of our first variables is w, so 2w, 3w...there is no w there at all, so it must be 0w; -2w.2240

Next, our x's: there are no x's in the first equation, so it is 0 x's; -2, -3, +5.2250

Next, 4y, 1y, 2y, 0y, -1z, 4z, 0z, 1z.2258

Great; now, we are looking to figure out what y is going to be.2270

We figure out that Ay is going to be the same thing, except it is going to have its y column swapped.2274

Notice that the y column is the third column in; so we are going to swap out the third column for the constants.2281

Other than that, it is going to look like our normal coefficient one.2287

So, we can copy over what we had in the previous one, except for that one column: 2, 3, 0, -2, 0, -2, -3, 5.2289

Now, this is the third column--this one right here--so we are swapping out for the column of constants.2301

That is 5, 16, 0, 17; and then back to copying the rest of it: -1, 4, 0, 1.2309

So, at this point, we have Ay; we have A; so we are going to need to figure out determinants.2317

First, let's figure out what the determinant of A is.2322

The determinant of A: if we want to figure out this here, remember: if we are going to be figuring out determinants,2324

we are going to be using cofactor expansion; so the very first thing we want to do is make a little +/- field: + - + -, - + - +, + - + -, - + - +.2331

We can use that as a reference point.2346

Which one would be the best one if we are looking to get the determinant of A?2348

If we are looking to get the determinant of A, which would be the best row or column to expand on?2354

I see two 0's on this column, so let's work off of that one.2358

The first 0 just disappears, because it is 0 times its cofactor; blow out that cofactor, because it is 0.2362

The next one is -3; so we are on the third row, second column, so that corresponds to that symbol, a negative.2368

It is negative, and then -3, what we have for what we are expanding around; negative -3 is minus -3;2376

and then times...what happens if we cut out everything on a line with that -3?2384

We have 2, 3, -2, 4, 1, 0, -1, 4, 1.2390

We have to keep going; now we are on the 2; let's swap to a new color.2400

2 here; we are on a + now, so it is +2 times...cut out what is on a line with that 2; so we have 2, 3, -2, 0, -2, 5, -1, 4, 1.2406

All right, at this point, we want to figure out what are the easiest rows or columns to expand on for these two matrices.2426

I notice that there is a 0 here and a 0 here; I personally find it easier to do expanding2433

based on a row than based on a column, so I will just choose to do rows.2440

- -3; these cancel to +3; so +3 times...expand on -2 first, so...2445

oh, and we are on a 3 x 3 now, so we are on that + there; so it is still positive...2452

it is 3 times...now we are figuring out the determinant of that matrix: -2 times...2457

cross out what is on a line with that: 4, -1, 1, 4; minus 0...but - 0 cancels out, so what is next after that?2462

Another plus: + 1 times...cross out what is on a line with that 1 here; 2, 4, 3, 1.2471

All right, let's work on our other half, the other determinant.2485

+2, times whatever the determinant is inside of this matrix: 2 here; 2 is a positive here,2489

because it is in the top left: so 2 times...whatever is on a line with that gets cancelled out.2495

So, we are left with -2, 4, 5, 1.2504

Then, 0 next: the 0 here we don't have to worry about.2508

And then, we are finally onto a +; but it is a negative 1, so + -1 times...cross out what is on a line with the -1; we are left with 3, -2, -2, 5.2511

OK, at this point, we just have a lot of arithmetic to work through.2525

3, -2...take the determinant of this matrix; we have 4 times 4; that is 16; 16 - -1 gets us +17.2530

Plus 1 times...forget about the 1...2 times 1 is 2, minus 3 times 4 is 12; so 2 - 12 is -10.2540

Plus 2 times...-2 times 1 is -2; minus 5 times 4 (is 20); so -2 - 20 is -22.2550

Plus -1...let's make it a negative...times...3 times 5 is 15; minus...-2 times -2 is +4; so 3 times 5 is 15, plus 4 is 19.2560

If you have difficulty doing that in your head, just write out the 2 x 2, as well.2577

Keep working through this: 3 times...-2 times 17 comes out to be -34; -34 - 10 + 2 times -22 (is -44), minus 19;2582

3 times -44 + 2(-44) - 19 becomes...oops, a mistake was made...oh, here it is.2602

I just caught my mistake--see how easy it is to make mistakes here?2622

That should be an important point: be really, really careful with this; it is really easy to make mistakes.2625

3 times 5 is 15, minus...-2 times -2 (let's work this one out carefully)...3 times 5, minus -2 times -2...2629

well, these cancel out, and we are left with +4; so 15 - 4 becomes 11.2640

So, this shouldn't be 19; it should be 11.2644

This shouldn't be a 19 here, either; it should be 11.2649

So, -44 - 11 becomes -55; see how easy it is to make mistakes?2652

I make mistakes; it is really easy to make mistakes; be very careful with this sort of stuff.2658

It is really, really a sad way to end up missing things, when you understand what is going on, but it is just one little, tiny arithmetic error.2662

All right, let's finish this one out.2670

3 times -44 becomes -132; plus 2 times -55 becomes -110; we combine those together, and we get -242.2671

-242 is the determinant of A; it is equal to -242; it takes a while to work through, doesn't it?2691

All right, the next one: We figured out the determinant of A; that comes out to be -242.2701

So, to use Cramer's Rule, we know that y is going to be equal to the determinant of Ay, over the determinant of A.2706

We now need to figure out what Ay comes out to be.2716

The determinant of Ay...let's figure this out.2720

We work through this one; I notice this nice row right here--we have three 0's on it.2727

That is going to make it easy to work through; if we are doing a cofactor expansion, we want to make our sign table.2733

OK, we can work along with that.2744

Our first one is a + on 0, but that doesn't matter; the next one is a - on -3.2745

- -3 on...we cut out what is on a line with that -3; we have 2, 3, -2, 5, 16, 17, -1, 4, 1.2752

Next is 0; once again, we don't have to worry about that.2772

Next is 0; once again, we don't have to worry about that.2774

All right, so we see that these cancel out, and we have 3 times...now we need to choose what we are going to expand along--which row or column.2777

Personally, I like the top row; I like expanding along rows, and 2, 5, -1 does at least have some kind of small numbers.2786

So, I will expand across that, just because I feel like it.2793

2, 5, -1: we will do it in three different colors here.2797

2 corresponds to this, so it is a positive 2, times what cuts along this...we are left with 16, 17, 4, 1.2801

The next one (do it with green): that corresponds to a negative there, so that is minus 5;2814

what does it cut out? We are left with 3, -2, 4, 1.2822

And then finally, go back to red; -1 corresponds to a positive, so + -1 times...what does it cross out? We have 3, -2, 16, 17 left.2831

All right, let's work this out: we have 3 times all of this stuff; 2 times...16 times 1 is 16; minus 17 times 4...2850

17 times 4 comes out to be -68; the next one: minus 5 times...3 times 1 is 32860

(after that mistake last time, let's be careful) minus...-2 times 4 is -8; so that will cancel out to plus.2874

Finally, we turn this to a minus, since it was times -1; 3 times 17 comes to 51; minus -2 times 16 becomes -32.2882

OK, keep working this out: 3 times 2 times 16 - 68 is -52,2899

minus 5 times 3 + 8 is 11, minus 51 - -32 becomes 51 + 32; 51 + 32 is - 83.2911

So, 3 times...2 times -52 becomes -104; minus 5 times 11 becomes -55; and still, minus 83.2929

We combine all of those together, and that gets us 3 times -242.2941

Now, you could go through and multiply this together, and you would get a number out of it.2948

But notice: we have -242 here, and later on, in just a few moments, we are about to divide by that previous detA at -242.2953

So, why don't we just leave this as 3 times -242; that is equal to the determinant of Ay, our special matrix for Ay.2962

At this point, we know from Cramer's Rule that y equals (cut out a little space for it)2976

the determinant of its special matrix, Ay, divided by the determinant of the coefficient matrix.2983

We figured out that the determinant of our special matrix is 3(-242), so 3(-242) divided by the determinant2991

of our coefficient matrix--that is also -242; -242/-242--those parts cancel out, and we are left with 3; so y = 3.3002

Sadly, there is no good way to check it at this point, if we are going to have to work through the whole thing,3013

because we would have to solve for each one of them, w and x and z.3018

On the bright side, solving for w, x, and z is only having to figure out the determinant of Aw, Ax, and Az,3023

because we have already figured out the determinant of A.3030

But still, it clearly takes some effort to take the determinants of even just a size 4 x 4, so it is pretty difficult.3031

However, if you have a graphing calculator, it would be pretty easy to go through and enter the matrix,3036

and then enter an augmented matrix, including the constants, and then get the reduced row echelon form3042

and see if y = 3 pops out as the answer that you would have from it.3049

It would be the case that that is what you would get out of it.3052

Or you could use Cramer's Rule and do determinants: figure out what Ax is; figure out what Aw is;3055

figure out what Az is; and then, be able to plug them all in and check afterwards.3061

Or you could also go through and do it with inverse matrices and see if y comes out to be 3, once you have figured out that on your calculator.3066

You can do this stuff by hand if you have to do it on a test;3073

but then, you can also, if you are allowed to just use the calculator (you just have to show your work)...3075

you can check your work in a second, different way to make sure that your work did come out to be true.3079

So, you can definitely get the problem right.3083

All right, the final example: do you remember that monster from solving systems of linear equations?3086

It is back, and we are going to solve it: we are going to knock out this thing that was way too difficult for us then.3090

It is going to be really easy for us now, because we have access to how inverse matrices work.3095

We can use calculators to be able to calculate an inverse matrix very quickly; this thing is going to be easy.3099

Our plan: remember, the idea was that we have the coefficient matrix A, times the column of the variables, is equal to the column of the constants.3104

So, AX = B: if we can figure out what A-1 is, we can multiply by A-1 on the left side on both cases.3117

A-1 cancels out there, and we are left with X = A-1B.3125

We already know what B is: B is this thing right here, so that part is pretty easy.3134

Can we figure out what A-1 is?3139

Well, this is A; I am assuming that we have access to a graphing calculator or some way to do matrix calculations.3141

Once again, matrix calculations are easy to do, but really tedious.3149

They take all of this time; it is easy to make a mistake, because just doing 100 calculations, you tend to make a mistake somewhere.3153

But that is what calculators and computers are for; that is why humans invented those sorts of things--3159

to be able to make tedious calculations like that go away, where we can trust the calculator3164

to do the number-crunching part, and we can trust us to do the thinking part (hopefully).3168

We figure what A is; it is going to be a big one: our u's first: 1u, -4u, 1u, -2, 1/5u, 2u;3174

next, our v's: 2v, 2v, 1v, 1/2v, -3v, +4v; 7w, 1w, 0w (because it didn't show up), 3w, -1w, -1w;3188

-3x, 1/3x, 0x, 0x, 2x, -3x; 4y, 2y, 1y, 2y, -1y, 5y; 2z, 1z, 1z, 4z, 4z, 0z.3207

So, what you do is: you take A, and you enter that into your graphing calculator.3225

You put that into a graphing calculator; you put that into some sort of matrix calculator.3230

You enter this into a calculator, or a computer, or something that is able to work with matrices.3233

Lots of programs are, because matrices are very useful.3243

Once again, we aren't even beginning to scratch the surface of how useful they are; we are just getting some sense with this one problem.3245

So, we enter this whole thing into a calculator; then you tell the calculator to take the inverse.3251

So, we do that; and I want to point out, before we actually go on to talk about the inverse:3257

you tell the calculator to take the inverse; before you do that, double-check that you entered the matrix correctly.3261

If you entered this wrong--if you entered this A, 6 x 6...that is 36 numbers that you just put into your calculator.3268

Chances are that you might have accidentally entered one of them wrong.3274

If you enter one of them wrong, your entire answer is going to be wrong.3277

Chances are it will end up being this awful decimal number, so you will think,3280

"Well, my teacher probably didn't give me something that would come out to be an awful decimal number."3283

But if you are working with something like physics, where you don't already know what the answer is going to be,3286

it is up to you to make sure that you get it in correctly the first time.3290

So, double-check: if you are entering a very large matrix, make certain that you entered that matrix correctly.3294

We have the entire matrix set up in our calculator, and we have double-checked that it is correct.3300

Now, we punch out A-1: on most calculators, that is going to end up being: take the matrix and raise it to the -1.3304

What does it come out to be; it comes out to be really ugly--it is awful.3310

For example, the very first term is going to be 1780/14131; the first row, second column, would be 45/14131; the third...this is awful.3315

So, what are we going to end up doing?3337

Do we have to write the whole thing down?3339

No, we don't have to write the whole thing down--it is in our calculator.3340

We just tell the calculator A-1, and then we don't have to worry about A-1 at all.3342

We don't have to figure it out and write the whole thing down on paper; there is no need for it.3348

The calculator will keep track of what the numbers for A-1 are,3352

because all we are concerned about is taking A-1 and applying it against B.3356

We leave it in the calculator; we know that X is going to be equal to A-1 times B.3362

All right, that is what we just figured out from our plan of thinking about this.3370

So, we have, in our calculator, that A-1 is in there.3374

We have it in the calculator; we don't have to actually see what the whole thing is, because it is already there.3378

What is our B? We enter in the column matrix, 41, 39, 4, 23, -30, 44; we make sure that our A-1 is multiplying from the left side.3383

Otherwise, it won't work at all.3396

And what does this end up coming out to be?3398

This comes out to be the deliciously simple -5, 4, 1, -3, 6, -1.3400

So, we just figured out that our X (all of our variables at once) is equal to...what were all of our variables?3410

It was u, and then we put in v, and then we put in w, and then we put in x, y, z.3420

So, they go in that order in our column: u, v, w, x, y, z = this thing that we just punched out, -5, 4, 1, -3, 6, -1.3425

So, u = -5; v = 4; w = 1; x = -3; y = 6; z = -1.3443

If we really wanted to at this point, we could check it; we could plug each one of these into any one of these equations;3450

and if it came out right, chances are that we probably got the entire thing right.3455

So, it might not be a bad idea to check at that point.3458

But also, as long as we were really careful with entering in our A, and careful with entering in our column of constants, our B,3460

everything should have worked out fine there; otherwise there is some other error that cropped up.3467

So, it becomes really, really easy, with just a little bit of thinking, and this calculator3470

(to take care of the awful manual work of the numbers, of just having to work through that many numbers)--3475

as long as we have the calculator to be able to do that part, so that it is quick and easy,3482

and we can trust that it came out right, and we are able to do the thought of what is going on,3485

we see that A-1, our coefficient matrix inverted, times what the equations come out to be,3489

our constant column matrix, just comes out to be the answers for each one of them.3496

It is really cool, really fast, and really easy; any time you have a large linear system,3500

or even a small linear system, and you just want to check it, you can have it done like that,3506

if you have access to a matrix calculator--pretty cool.3509

All right, we will see you at Educator.com later--goodbye!3512