# Linear Algebra Linear Systems

Section 1: Linear Equations and Matrices: Lecture 1 | 39:03 min

In this introduction to linear algebra we want to start you off with some basics of the course. You should have already been introduced to equations with one variable; what linear algebra utilizes greatly are multiple equations with multiple variables. However, there are not always specific answers for each of the variables, and so we’ll also discussion the types of solutions you’ll encounter when doing some problems. We’ll also equip you with a solving method to be used should there be exactly one solution for each variable. Next video we’ll get into a new way to think of data rather than a bunch of variables.

Table of Contents

39m 3s

- Intro0:00
- Linear Systems1:20
- Introduction to Linear Systems1:21
- Examples10:35
- Example 110:36
- Example 213:44
- Example 316:12
- Example 423:48
- Example 528:23
- Example 632:32
- Number of Solutions35:08
- One Solution, No Solution, Infinitely Many Solutions35:09
- Method of Elimination36:57
- Method of Elimination36:58

30m 34s

- Intro0:00
- Matrices0:47
- Definition and Example of Matrices0:48
- Square Matrix7:55
- Diagonal Matrix9:31
- Operations with Matrices10:35
- Matrix Addition10:36
- Scalar Multiplication15:01
- Transpose of a Matrix17:51
- Matrix Types23:17
- Regular: m x n Matrix of m Rows and n Column23:18
- Square: n x n Matrix With an Equal Number of Rows and Columns23:44
- Diagonal: A Square Matrix Where All Entries OFF the Main Diagonal are '0'24:07
- Matrix Operations24:37
- Matrix Operations24:38
- Example25:55
- Example25:56

41m 42s

- Intro0:00
- Dot Product1:04
- Example of Dot Product1:05
- Matrix Multiplication7:05
- Definition7:06
- Example 112:26
- Example 217:38
- Matrices and Linear Systems21:24
- Matrices and Linear Systems21:25
- Example 129:56
- Example 232:30
- Summary33:56
- Dot Product of Two Vectors and Matrix Multiplication33:57
- Summary, cont.35:06
- Matrix Representations of Linear Systems35:07
- Examples35:34
- Examples35:35

43m 17s

- Intro0:00
- Properties of Addition1:11
- Properties of Addition: A1:12
- Properties of Addition: B2:30
- Properties of Addition: C2:57
- Properties of Addition: D4:20
- Properties of Addition5:22
- Properties of Addition5:23
- Properties of Multiplication6:47
- Properties of Multiplication: A7:46
- Properties of Multiplication: B8:13
- Properties of Multiplication: C9:18
- Example: Properties of Multiplication9:35
- Definitions and Properties (Multiplication)14:02
- Identity Matrix: n x n matrix14:03
- Let A Be a Matrix of m x n15:23
- Definitions and Properties (Multiplication)18:36
- Definitions and Properties (Multiplication)18:37
- Properties of Scalar Multiplication22:54
- Properties of Scalar Multiplication: A23:39
- Properties of Scalar Multiplication: B24:04
- Properties of Scalar Multiplication: C24:29
- Properties of Scalar Multiplication: D24:48
- Properties of the Transpose25:30
- Properties of the Transpose25:31
- Properties of the Transpose30:28
- Example30:29
- Properties of Matrix Addition33:25
- Let A, B, C, and D Be m x n Matrices33:26
- There is a Unique m x n Matrix, 0, Such That…33:48
- Unique Matrix D34:17
- Properties of Matrix Multiplication34:58
- Let A, B, and C Be Matrices of the Appropriate Size34:59
- Let A Be Square Matrix (n x n)35:44
- Properties of Scalar Multiplication36:35
- Let r and s Be Real Numbers, and A and B Matrices36:36
- Properties of the Transpose37:10
- Let r Be a Scalar, and A and B Matrices37:12
- Example37:58
- Example37:59

38m 14s

- Intro0:00
- Reduced Row Echelon Form0:29
- An m x n Matrix is in Reduced Row Echelon Form If:0:30
- Reduced Row Echelon Form2:58
- Example: Reduced Row Echelon Form2:59
- Theorem8:30
- Every m x n Matrix is Row-Equivalent to a UNIQUE Matrix in Reduced Row Echelon Form8:31
- Systematic and Careful Example10:02
- Step 110:54
- Step 211:33
- Step 312:50
- Step 414:02
- Step 515:31
- Step 617:28
- Example30:39
- Find the Reduced Row Echelon Form of a Given m x n Matrix30:40

28m 54s

- Intro0:00
- Solutions of Linear Systems0:11
- Solutions of Linear Systems0:13
- Example I3:25
- Solve the Linear System 13:26
- Solve the Linear System 214:31
- Example II17:41
- Solve the Linear System 317:42
- Solve the Linear System 420:17
- Homogeneous Systems21:54
- Homogeneous Systems Overview21:55
- Theorem and Example24:01

40m 10s

- Intro0:00
- Finding the Inverse of a Matrix0:41
- Finding the Inverse of a Matrix0:42
- Properties of Non-Singular Matrices6:38
- Practical Procedure9:15
- Step19:16
- Step 210:10
- Step 310:46
- Example: Finding Inverse12:50
- Linear Systems and Inverses17:01
- Linear Systems and Inverses17:02
- Theorem and Example21:15
- Theorem26:32
- Theorem26:33
- List of Non-Singular Equivalences28:37
- Example: Does the Following System Have a Non-trivial Solution?30:13
- Example: Inverse of a Matrix36:16

21m 25s

- Intro0:00
- Determinants0:37
- Introduction to Determinants0:38
- Example6:12
- Properties9:00
- Properties 1-59:01
- Example10:14
- Properties, cont.12:28
- Properties 6 & 712:29
- Example14:14
- Properties, cont.18:34
- Properties 8 & 918:35
- Example19:21

59m 31s

- Intro0:00
- Cofactor Expansions and Their Application0:42
- Cofactor Expansions and Their Application0:43
- Example 13:52
- Example 27:08
- Evaluation of Determinants by Cofactor9:38
- Theorem9:40
- Example 111:41
- Inverse of a Matrix by Cofactor22:42
- Inverse of a Matrix by Cofactor and Example22:43
- More Example36:22
- List of Non-Singular Equivalences43:07
- List of Non-Singular Equivalences43:08
- Example44:38
- Cramer's Rule52:22
- Introduction to Cramer's Rule and Example52:23

46m 54s

- Intro0:00
- Vectors in the Plane0:38
- Vectors in the Plane0:39
- Example 18:25
- Example 215:23
- Vector Addition and Scalar Multiplication19:33
- Vector Addition19:34
- Scalar Multiplication24:08
- Example26:25
- The Angle Between Two Vectors29:33
- The Angle Between Two Vectors29:34
- Example33:54
- Properties of the Dot Product and Unit Vectors38:17
- Properties of the Dot Product and Unit Vectors38:18
- Defining Unit Vectors40:01
- 2 Very Important Unit Vectors41:56

52m 44s

- Intro0:00
- n-Vectors0:58
- 4-Vector0:59
- 7-Vector1:50
- Vector Addition2:43
- Scalar Multiplication3:37
- Theorem: Part 14:24
- Theorem: Part 211:38
- Right and Left Handed Coordinate System14:19
- Projection of a Point Onto a Coordinate Line/Plane17:20
- Example21:27
- Cauchy-Schwarz Inequality24:56
- Triangle Inequality36:29
- Unit Vector40:34
- Vectors and Dot Products44:23
- Orthogonal Vectors44:24
- Cauchy-Schwarz Inequality45:04
- Triangle Inequality45:21
- Example 145:40
- Example 248:16

48m 53s

- Intro0:00
- Introduction to Linear Transformations0:44
- Introduction to Linear Transformations0:45
- Example 19:01
- Example 211:33
- Definition of Linear Mapping14:13
- Example 322:31
- Example 426:07
- Example 530:36
- Examples36:12
- Projection Mapping36:13
- Images, Range, and Linear Mapping38:33
- Example of Linear Transformation42:02

34m 8s

- Intro0:00
- Linear Transformations1:29
- Linear Transformations1:30
- Theorem 17:15
- Theorem 29:20
- Example 1: Find L (-3, 4, 2)11:17
- Example 2: Is It Linear?17:11
- Theorem 325:57
- Example 3: Finding the Standard Matrix29:09

37m 54s

- Intro0:00
- Lines and Plane0:36
- Example 10:37
- Example 27:07
- Lines in IR39:53
- Parametric Equations14:58
- Example 317:26
- Example 420:11
- Planes in IR325:19
- Example 531:12
- Example 634:18

42m 19s

- Intro0:00
- Vector Spaces3:43
- Definition of Vector Spaces3:44
- Vector Spaces 15:19
- Vector Spaces 29:34
- Real Vector Space and Complex Vector Space14:01
- Example 115:59
- Example 218:42
- Examples26:22
- More Examples26:23
- Properties of Vector Spaces32:53
- Properties of Vector Spaces Overview32:54
- Property A34:31
- Property B36:09
- Property C36:38
- Property D37:54
- Property F39:00

43m 37s

- Intro0:00
- Subspaces0:47
- Defining Subspaces0:48
- Example 13:08
- Example 23:49
- Theorem7:26
- Example 39:11
- Example 412:30
- Example 516:05
- Linear Combinations23:27
- Definition 123:28
- Example 125:24
- Definition 229:49
- Example 231:34
- Theorem32:42
- Example 334:00

33m 15s

- Intro0:00
- A Spanning Set for a Vector Space1:10
- A Spanning Set for a Vector Space1:11
- Procedure to Check if a Set of Vectors Spans a Vector Space3:38
- Example 16:50
- Example 214:28
- Example 321:06
- Example 422:15

17m 20s

- Intro0:00
- Linear Independence0:32
- Definition0:39
- Meaning3:00
- Procedure for Determining if a Given List of Vectors is Linear Independence or Linear Dependence5:00
- Example 17:21
- Example 210:20

31m 20s

- Intro0:00
- Basis and Dimension0:23
- Definition0:24
- Example 13:30
- Example 2: Part A4:00
- Example 2: Part B6:53
- Theorem 19:40
- Theorem 211:32
- Procedure for Finding a Subset of S that is a Basis for Span S14:20
- Example 316:38
- Theorem 321:08
- Example 425:27

24m 45s

- Intro0:00
- Homogeneous Systems0:51
- Homogeneous Systems0:52
- Procedure for Finding a Basis for the Null Space of Ax = 02:56
- Example 17:39
- Example 218:03
- Relationship Between Homogeneous and Non-Homogeneous Systems19:47

35m 3s

- Intro0:00
- Rank of a Matrix1:47
- Definition1:48
- Theorem 18:14
- Example 19:38
- Defining Row and Column Rank16:53
- If We Want a Basis for Span S Consisting of Vectors From S22:00
- If We want a Basis for Span S Consisting of Vectors Not Necessarily in S24:07
- Example 2: Part A26:44
- Example 2: Part B32:10

29m 26s

- Intro0:00
- Rank of a Matrix0:17
- Example 1: Part A0:18
- Example 1: Part B5:58
- Rank of a Matrix Review: Rows, Columns, and Row Rank8:22
- Procedure for Computing the Rank of a Matrix14:36
- Theorem 1: Rank + Nullity = n16:19
- Example 217:48
- Rank & Singularity20:09
- Example 321:08
- Theorem 223:25
- List of Non-Singular Equivalences24:24
- List of Non-Singular Equivalences24:25

27m 3s

- Intro0:00
- Coordinates of a Vector1:07
- Coordinates of a Vector1:08
- Example 18:35
- Example 215:28
- Example 3: Part A19:15
- Example 3: Part B22:26

33m 47s

- Intro0:00
- Change of Basis & Transition Matrices0:56
- Change of Basis & Transition Matrices0:57
- Example 110:44
- Example 220:44
- Theorem23:37
- Example 3: Part A26:21
- Example 3: Part B32:05

32m 53s

- Intro0:00
- Orthonormal Bases in n-Space1:02
- Orthonormal Bases in n-Space: Definition1:03
- Example 14:31
- Theorem 16:55
- Theorem 28:00
- Theorem 39:04
- Example 210:07
- Theorem 213:54
- Procedure for Constructing an O/N Basis16:11
- Example 321:42

21m 27s

- Intro0:00
- Orthogonal Complements0:19
- Definition0:20
- Theorem 15:36
- Example 16:58
- Theorem 213:26
- Theorem 315:06
- Example 218:20

33m 49s

- Intro0:00
- Relations Among the Four Fundamental Vector Spaces Associated with a Matrix A2:16
- Four Spaces Associated With A (If A is m x n)2:17
- Theorem4:49
- Example 17:17
- Null Space and Column Space10:48
- Projections and Applications16:50
- Projections and Applications16:51
- Projection Illustration21:00
- Example 123:51
- Projection Illustration Review30:15

38m 11s

- Intro0:00
- Eigenvalues and Eigenvectors0:38
- Eigenvalues and Eigenvectors0:39
- Definition 13:30
- Example 17:20
- Example 210:19
- Definition 221:15
- Example 323:41
- Theorem 126:32
- Theorem 227:56
- Example 429:14
- Review34:32

29m 55s

- Intro0:00
- Similar Matrices and Diagonalization0:25
- Definition 10:26
- Example 12:00
- Properties3:38
- Definition 24:57
- Theorem 16:12
- Example 39:37
- Theorem 212:40
- Example 419:12
- Example 520:55
- Procedure for Diagonalizing Matrix A: Step 124:21
- Procedure for Diagonalizing Matrix A: Step 225:04
- Procedure for Diagonalizing Matrix A: Step 325:38
- Procedure for Diagonalizing Matrix A: Step 427:02

30m 14s

- Intro0:00
- Diagonalization of Symmetric Matrices1:15
- Diagonalization of Symmetric Matrices1:16
- Theorem 12:24
- Theorem 23:27
- Example 14:47
- Definition 16:44
- Example 28:15
- Theorem 310:28
- Theorem 412:31
- Example 318:00

24m 5s

- Intro0:00
- Linear Mappings2:08
- Definition2:09
- Linear Operator7:36
- Projection8:48
- Dilation9:40
- Contraction10:07
- Reflection10:26
- Rotation11:06
- Example 113:00
- Theorem 118:16
- Theorem 219:20

26m 38s

- Intro0:00
- Kernel and Range of a Linear Map0:28
- Definition 10:29
- Example 14:36
- Example 28:12
- Definition 210:34
- Example 313:34
- Theorem 116:01
- Theorem 218:26
- Definition 321:11
- Theorem 324:28

25m 54s

- Intro0:00
- Kernel and Range of a Linear Map1:39
- Theorem 11:40
- Example 1: Part A2:32
- Example 1: Part B8:12
- Example 1: Part C13:11
- Example 1: Part D14:55
- Theorem 216:50
- Theorem 323:00

33m 21s

- Intro0:00
- Matrix of a Linear Map0:11
- Theorem 11:24
- Procedure for Computing to Matrix: Step 17:10
- Procedure for Computing to Matrix: Step 28:58
- Procedure for Computing to Matrix: Step 39:50
- Matrix of a Linear Map: Property10:41
- Example 114:07
- Example 218:12
- Example 324:31

For more information, please see full course syllabus of Linear Algebra

0 answers

Post by John Lins on January 30, 2017

Could you please explain what would be the solution to this question below?

Show that this pair of augmented matrices are row equivalent, assuming ad-be# 0:

(a b |e) ~ (1 0 |(de-bf) / (ad-bc))

c d |f (0 1 |(af-ce) / (ad-bc)

1 answer

Last reply by: Professor Hovasapian

Wed Jan 18, 2017 8:13 PM

Post by Mohsin Alibrahim on January 10, 2017

Hi Professor H

In ex 5, why didn't you multiply the first equation by (-5) and eliminate the x variables the x immediately instead of what you did when worked on 1,2 and 1,4 separately.

1 answer

Last reply by: Professor Hovasapian

Tue Nov 29, 2016 2:40 AM

Post by manu vats on November 25, 2016

which textbook should i use for this course

1 answer

Last reply by: Professor Hovasapian

Sun Nov 6, 2016 4:37 PM

Post by El Einstein on November 5, 2016

Would you classify "infinite # of solutions" as Consistent or Inconsistent? Im slightly confused.

I'm assuming that as long as the system of linear equations has at least one solution, it will be considered as Consistent. Is this a correct assumption?

3 answers

Last reply by: cary pope

Fri Nov 18, 2016 9:27 PM

Post by cary pope on October 15, 2016

Professor Hovasapian, You are so good at explaining science and mathematical concepts. Your videos are well formed and extremely useful. The first lessons of yours I watched was on multivariable calculus. I was way ahead of my class and fell in love with vector calculus, with your videos and explanations being a major factor. I feel lucky, I'm taking linear algebra and chemistry at the moment and was pleasantly surprised to find out that you have courses for each on educator.com. I just wanted to mention how much you're helping me and how good you are. I love how you simplify things and include tricks to visualize the process of what you are actually doing with the mathematics, your multivariable calculus course was brilliant. I'm enjoying your chemistry and linear algebra coases as well. Any chances of you making a course on complex analysis? I'm taking that next semester!

Thank you for making these effective videos that are on target and to the purpose!

Cary

1 answer

Last reply by: Professor Hovasapian

Sat Sep 10, 2016 2:17 AM

Post by Kaye Lim on September 9, 2016

For a case of 2 unknowns with 3 equation, after solving this, let's say we get 1 solution (x,y). How would the graph look like? Does it mean 3 lines meet at a point (x,y) in the solution?

-Is it possible for these 3 lines to meet at 2 points?

1 answer

Last reply by: Professor Hovasapian

Fri Jun 17, 2016 6:24 PM

Post by Emily Lewis on June 16, 2016

Why did you decide to use elimination for all of these examples when substitution would have been much easier for some of them?

1 answer

Last reply by: Professor Hovasapian

Tue Apr 7, 2015 10:50 PM

Post by Micheal Bingham on March 31, 2015

You have an absolutely beautiful verbiage, I know that this does not pertain to the lecture but do you have any advice how one could become as articulate as you?

1 answer

Last reply by: Professor Hovasapian

Mon Mar 2, 2015 6:37 PM

Post by julius mogyorossy on March 1, 2015

Dr. do I need your course to take the college Algebra CLEP test?

1 answer

Last reply by: Professor Hovasapian

Mon Feb 2, 2015 4:02 PM

Post by Danial Shadmany on February 1, 2015

Hi Professor,

I was just wondering if you will cover singular value decomposition in this course. I didn't see it in the syllabus and wondered if you'd still cover it in this course.

Thanks!

1 answer

Last reply by: Professor Hovasapian

Fri Nov 21, 2014 9:29 PM

Post by Eric Liu on November 21, 2014

Hi Professor Hovasapian,

I know your Calculus AB course is going to be released sometime in the coming months, but I was wondering if you will be doing a Calculus BC course for educator.com as well?

Thanks!

2 answers

Last reply by: Miguel Villarreal

Mon Jun 2, 2014 9:44 AM

Post by Miguel Villarreal on June 2, 2014

@31:32 Example VI 3 equations and 2 unknowns because y=4 if the other y=x another variable would we solve for both or does become an inconsistent system because they are different?

Thank you

0 answers

Post by MOHAMMED ALHUMAIDI on December 7, 2013

Hello

When i try to download lecture slides it is show without any answer on it??

1 answer

Last reply by: Professor Hovasapian

Mon Sep 23, 2013 4:17 PM

Post by Jawad Mustafa on September 22, 2013

Thank you very much for offering this course

Jawad Mustafa

Amman - Jordan

3 answers

Last reply by: Professor Hovasapian

Wed Dec 30, 2015 12:03 AM

Post by Manfred Berger on May 29, 2013

I've been thinking about example 4 for a bit, and it seems to me that the reason you're not getting a single point as your solution there is you're intersecting 2 planes, which in turn leads to a line as your result.

1 answer

Last reply by: Professor Hovasapian

Fri Apr 12, 2013 4:33 PM

Post by Rishabh Jain on April 12, 2013

I love your lecture SIR !!! AMAZINGGGGGGGGG !!!!!!!!!!

1 answer

Last reply by: Professor Hovasapian

Thu Oct 11, 2012 3:29 PM

Post by Aniket Dhawan on October 11, 2012

You are a great teacher. I really liked your explanations,they helped me a lot.

Thankyou professor

3 answers

Last reply by: Aniket Dhawan

Thu Oct 11, 2012 5:09 AM

Post by Suhaib Hasan on October 4, 2012

Your comment about induction and math was great.

2 answers

Last reply by: Rob Lee

Thu Sep 27, 2012 6:18 PM

Post by robert lee on September 26, 2012

Question: at 36:02. I am slightly confused with your graph, I understand the no solution case, where the lines are parallel and never meet.

The one solution case meets at one point (in other words if it was three or more lines, they should all intersect at the same point then?)

Now the infinite case, I do not understand at all, what does it mean when the line is on top of another line? Doesn't this just mean that they meet at one point? So it would be just like the one solution case then... How is this possible?

In my imagination, an infinite solution would be more like a graph of a sin and cos equation, cause then they would intersect at multiple points, but then this would not be linear??

Can you please clarify?

Thank you.

1 answer

Last reply by: Professor Hovasapian

Fri Sep 21, 2012 2:34 PM

Post by Erdem Balikci on September 20, 2012

Very well done! Clear and smooth!!

0 answers

Post by Maimouna Louche on June 17, 2012

Thanks I get it now, it looked scary for nothing.

0 answers

Post by Maimouna Louche on June 15, 2012

I will be taking this class soon. Man it look hard :( I will be fine, I have Educator now! ^^

0 answers

Post by Real Schiran on February 29, 2012

All understood. This lecture is very clear. Thanks

1 answer

Last reply by: Constantin Ficiu

Thu Oct 24, 2013 2:34 PM

Post by amir szeinberg on February 18, 2012

I have a problem re-entering the lecture in the middle, say in the 20th minute, and I have to strart from the begining. Is there any thing I could do about it?

Thanks in advance,

Amir

0 answers

Post by thomas kotch on December 18, 2011

He is great!

0 answers

Post by Senghuot Lim on December 18, 2011

Einstein?

0 answers

Post by Arthur Bookstein on October 12, 2011

Very well explained.

1 answer

Last reply by: Constantin Ficiu

Thu Oct 24, 2013 2:30 PM

Post by Jason Mannion on October 4, 2011

I subscribed to this site because I was having great difficulties in my university Linear Algebra class. So I have only watched the "Linear Systems" lecture, and from that I can conclude a few things: 1)Dr. Hovasapian actually has the ability to organize a course, 2) He knows how to present material in a educational manner, and 3) in the one video I learned more than I did in several weeks of my university lectures.

(the problem is simply that my professor lacks any teaching abilities, and cannot organize the material. We started our course with complex numbers, and then went straight into vectors and subspaces, without any explanation as to what a linear system was, how to solve one, or even how to do basic matrix arithmetic! In fact, we never see any matrices in class!)

0 answers

Post by Travis Torres on October 1, 2011

Very well done. I'll admit I was a little confused by the initial lecture on Linear Systems, but the examples themselves were very informative and helpful.