Home » Mathematics » Linear Algebra
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• 34 Lessons (23hr : 32min)
• Audio: English
• English

Join Professor Raffi Hovasapian in his time-saving Linear Algebra online course that focuses on clear explanations with tons of step-by-step examples. Raffi brings even more enthusiasm as Linear Algebra is his favorite subject and he aims to make it understandable for all.

## Section 1: Linear Equations and Matrices

Linear Systems 39:03
Intro 0:00
Linear Systems 1:20
Introduction to Linear Systems 1:21
Examples 10:35
Example 1 10:36
Example 2 13:44
Example 3 16:12
Example 4 23:48
Example 5 28:23
Example 6 32:32
Number of Solutions 35:08
One Solution, No Solution, Infinitely Many Solutions 35:09
Method of Elimination 36:57
Method of Elimination 36:58
Matrices 30:34
Intro 0:00
Matrices 0:47
Definition and Example of Matrices 0:48
Square Matrix 7:55
Diagonal Matrix 9:31
Operations with Matrices 10:35
Scalar Multiplication 15:01
Transpose of a Matrix 17:51
Matrix Types 23:17
Regular: m x n Matrix of m Rows and n Column 23:18
Square: n x n Matrix With an Equal Number of Rows and Columns 23:44
Diagonal: A Square Matrix Where All Entries OFF the Main Diagonal are '0' 24:07
Matrix Operations 24:37
Matrix Operations 24:38
Example 25:55
Example 25:56
Dot Product & Matrix Multiplication 41:42
Intro 0:00
Dot Product 1:04
Example of Dot Product 1:05
Matrix Multiplication 7:05
Definition 7:06
Example 1 12:26
Example 2 17:38
Matrices and Linear Systems 21:24
Matrices and Linear Systems 21:25
Example 1 29:56
Example 2 32:30
Summary 33:56
Dot Product of Two Vectors and Matrix Multiplication 33:57
Summary, cont. 35:06
Matrix Representations of Linear Systems 35:07
Examples 35:34
Examples 35:35
Properties of Matrix Operation 43:17
Intro 0:00
Properties of Multiplication 6:47
Properties of Multiplication: A 7:46
Properties of Multiplication: B 8:13
Properties of Multiplication: C 9:18
Example: Properties of Multiplication 9:35
Definitions and Properties (Multiplication) 14:02
Identity Matrix: n x n matrix 14:03
Let A Be a Matrix of m x n 15:23
Definitions and Properties (Multiplication) 18:36
Definitions and Properties (Multiplication) 18:37
Properties of Scalar Multiplication 22:54
Properties of Scalar Multiplication: A 23:39
Properties of Scalar Multiplication: B 24:04
Properties of Scalar Multiplication: C 24:29
Properties of Scalar Multiplication: D 24:48
Properties of the Transpose 25:30
Properties of the Transpose 25:31
Properties of the Transpose 30:28
Example 30:29
Let A, B, C, and D Be m x n Matrices 33:26
There is a Unique m x n Matrix, 0, Such That… 33:48
Unique Matrix D 34:17
Properties of Matrix Multiplication 34:58
Let A, B, and C Be Matrices of the Appropriate Size 34:59
Let A Be Square Matrix (n x n) 35:44
Properties of Scalar Multiplication 36:35
Let r and s Be Real Numbers, and A and B Matrices 36:36
Properties of the Transpose 37:10
Let r Be a Scalar, and A and B Matrices 37:12
Example 37:58
Example 37:59
Solutions of Linear Systems, Part 1 38:14
Intro 0:00
Reduced Row Echelon Form 0:29
An m x n Matrix is in Reduced Row Echelon Form If: 0:30
Reduced Row Echelon Form 2:58
Example: Reduced Row Echelon Form 2:59
Theorem 8:30
Every m x n Matrix is Row-Equivalent to a UNIQUE Matrix in Reduced Row Echelon Form 8:31
Systematic and Careful Example 10:02
Step 1 10:54
Step 2 11:33
Step 3 12:50
Step 4 14:02
Step 5 15:31
Step 6 17:28
Example 30:39
Find the Reduced Row Echelon Form of a Given m x n Matrix 30:40
Solutions of Linear Systems, Part II 28:54
Intro 0:00
Solutions of Linear Systems 0:11
Solutions of Linear Systems 0:13
Example I 3:25
Solve the Linear System 1 3:26
Solve the Linear System 2 14:31
Example II 17:41
Solve the Linear System 3 17:42
Solve the Linear System 4 20:17
Homogeneous Systems 21:54
Homogeneous Systems Overview 21:55
Theorem and Example 24:01
Inverse of a Matrix 40:10
Intro 0:00
Finding the Inverse of a Matrix 0:41
Finding the Inverse of a Matrix 0:42
Properties of Non-Singular Matrices 6:38
Practical Procedure 9:15
Step1 9:16
Step 2 10:10
Step 3 10:46
Example: Finding Inverse 12:50
Linear Systems and Inverses 17:01
Linear Systems and Inverses 17:02
Theorem and Example 21:15
Theorem 26:32
Theorem 26:33
List of Non-Singular Equivalences 28:37
Example: Does the Following System Have a Non-trivial Solution? 30:13
Example: Inverse of a Matrix 36:16

## Section 2: Determinants

Determinants 21:25
Intro 0:00
Determinants 0:37
Introduction to Determinants 0:38
Example 6:12
Properties 9:00
Properties 1-5 9:01
Example 10:14
Properties, cont. 12:28
Properties 6 & 7 12:29
Example 14:14
Properties, cont. 18:34
Properties 8 & 9 18:35
Example 19:21
Cofactor Expansions 59:31
Intro 0:00
Cofactor Expansions and Their Application 0:42
Cofactor Expansions and Their Application 0:43
Example 1 3:52
Example 2 7:08
Evaluation of Determinants by Cofactor 9:38
Theorem 9:40
Example 1 11:41
Inverse of a Matrix by Cofactor 22:42
Inverse of a Matrix by Cofactor and Example 22:43
More Example 36:22
List of Non-Singular Equivalences 43:07
List of Non-Singular Equivalences 43:08
Example 44:38
Cramer's Rule 52:22
Introduction to Cramer's Rule and Example 52:23

## Section 3: Vectors in Rn

Vectors in the Plane 46:54
Intro 0:00
Vectors in the Plane 0:38
Vectors in the Plane 0:39
Example 1 8:25
Example 2 15:23
Vector Addition and Scalar Multiplication 19:33
Scalar Multiplication 24:08
Example 26:25
The Angle Between Two Vectors 29:33
The Angle Between Two Vectors 29:34
Example 33:54
Properties of the Dot Product and Unit Vectors 38:17
Properties of the Dot Product and Unit Vectors 38:18
Defining Unit Vectors 40:01
2 Very Important Unit Vectors 41:56
n-Vector 52:44
Intro 0:00
n-Vectors 0:58
4-Vector 0:59
7-Vector 1:50
Scalar Multiplication 3:37
Theorem: Part 1 4:24
Theorem: Part 2 11:38
Right and Left Handed Coordinate System 14:19
Projection of a Point Onto a Coordinate Line/Plane 17:20
Example 21:27
Cauchy-Schwarz Inequality 24:56
Triangle Inequality 36:29
Unit Vector 40:34
Vectors and Dot Products 44:23
Orthogonal Vectors 44:24
Cauchy-Schwarz Inequality 45:04
Triangle Inequality 45:21
Example 1 45:40
Example 2 48:16
Linear Transformation 48:53
Intro 0:00
Introduction to Linear Transformations 0:44
Introduction to Linear Transformations 0:45
Example 1 9:01
Example 2 11:33
Definition of Linear Mapping 14:13
Example 3 22:31
Example 4 26:07
Example 5 30:36
Examples 36:12
Projection Mapping 36:13
Images, Range, and Linear Mapping 38:33
Example of Linear Transformation 42:02
Linear Transformations, Part II 34:08
Intro 0:00
Linear Transformations 1:29
Linear Transformations 1:30
Theorem 1 7:15
Theorem 2 9:20
Example 1: Find L (-3, 4, 2) 11:17
Example 2: Is It Linear? 17:11
Theorem 3 25:57
Example 3: Finding the Standard Matrix 29:09
Lines and Planes 37:54
Intro 0:00
Lines and Plane 0:36
Example 1 0:37
Example 2 7:07
Lines in IR3 9:53
Parametric Equations 14:58
Example 3 17:26
Example 4 20:11
Planes in IR3 25:19
Example 5 31:12
Example 6 34:18

## Section 4: Real Vector Spaces

Vector Spaces 42:19
Intro 0:00
Vector Spaces 3:43
Definition of Vector Spaces 3:44
Vector Spaces 1 5:19
Vector Spaces 2 9:34
Real Vector Space and Complex Vector Space 14:01
Example 1 15:59
Example 2 18:42
Examples 26:22
More Examples 26:23
Properties of Vector Spaces 32:53
Properties of Vector Spaces Overview 32:54
Property A 34:31
Property B 36:09
Property C 36:38
Property D 37:54
Property F 39:00
Subspaces 43:37
Intro 0:00
Subspaces 0:47
Defining Subspaces 0:48
Example 1 3:08
Example 2 3:49
Theorem 7:26
Example 3 9:11
Example 4 12:30
Example 5 16:05
Linear Combinations 23:27
Definition 1 23:28
Example 1 25:24
Definition 2 29:49
Example 2 31:34
Theorem 32:42
Example 3 34:00
Spanning Set for a Vector Space 33:15
Intro 0:00
A Spanning Set for a Vector Space 1:10
A Spanning Set for a Vector Space 1:11
Procedure to Check if a Set of Vectors Spans a Vector Space 3:38
Example 1 6:50
Example 2 14:28
Example 3 21:06
Example 4 22:15
Linear Independence 17:20
Intro 0:00
Linear Independence 0:32
Definition 0:39
Meaning 3:00
Procedure for Determining if a Given List of Vectors is Linear Independence or Linear Dependence 5:00
Example 1 7:21
Example 2 10:20
Basis & Dimension 31:20
Intro 0:00
Basis and Dimension 0:23
Definition 0:24
Example 1 3:30
Example 2: Part A 4:00
Example 2: Part B 6:53
Theorem 1 9:40
Theorem 2 11:32
Procedure for Finding a Subset of S that is a Basis for Span S 14:20
Example 3 16:38
Theorem 3 21:08
Example 4 25:27
Homogeneous Systems 24:45
Intro 0:00
Homogeneous Systems 0:51
Homogeneous Systems 0:52
Procedure for Finding a Basis for the Null Space of Ax = 0 2:56
Example 1 7:39
Example 2 18:03
Relationship Between Homogeneous and Non-Homogeneous Systems 19:47
Rank of a Matrix, Part I 35:03
Intro 0:00
Rank of a Matrix 1:47
Definition 1:48
Theorem 1 8:14
Example 1 9:38
Defining Row and Column Rank 16:53
If We Want a Basis for Span S Consisting of Vectors From S 22:00
If We want a Basis for Span S Consisting of Vectors Not Necessarily in S 24:07
Example 2: Part A 26:44
Example 2: Part B 32:10
Rank of a Matrix, Part II 29:26
Intro 0:00
Rank of a Matrix 0:17
Example 1: Part A 0:18
Example 1: Part B 5:58
Rank of a Matrix Review: Rows, Columns, and Row Rank 8:22
Procedure for Computing the Rank of a Matrix 14:36
Theorem 1: Rank + Nullity = n 16:19
Example 2 17:48
Rank & Singularity 20:09
Example 3 21:08
Theorem 2 23:25
List of Non-Singular Equivalences 24:24
List of Non-Singular Equivalences 24:25
Coordinates of a Vector 27:03
Intro 0:00
Coordinates of a Vector 1:07
Coordinates of a Vector 1:08
Example 1 8:35
Example 2 15:28
Example 3: Part A 19:15
Example 3: Part B 22:26
Change of Basis & Transition Matrices 33:47
Intro 0:00
Change of Basis & Transition Matrices 0:56
Change of Basis & Transition Matrices 0:57
Example 1 10:44
Example 2 20:44
Theorem 23:37
Example 3: Part A 26:21
Example 3: Part B 32:05
Orthonormal Bases in n-Space 32:53
Intro 0:00
Orthonormal Bases in n-Space 1:02
Orthonormal Bases in n-Space: Definition 1:03
Example 1 4:31
Theorem 1 6:55
Theorem 2 8:00
Theorem 3 9:04
Example 2 10:07
Theorem 2 13:54
Procedure for Constructing an O/N Basis 16:11
Example 3 21:42
Orthogonal Complements, Part I 21:27
Intro 0:00
Orthogonal Complements 0:19
Definition 0:20
Theorem 1 5:36
Example 1 6:58
Theorem 2 13:26
Theorem 3 15:06
Example 2 18:20
Orthogonal Complements, Part II 33:49
Intro 0:00
Relations Among the Four Fundamental Vector Spaces Associated with a Matrix A 2:16
Four Spaces Associated With A (If A is m x n) 2:17
Theorem 4:49
Example 1 7:17
Null Space and Column Space 10:48
Projections and Applications 16:50
Projections and Applications 16:51
Projection Illustration 21:00
Example 1 23:51
Projection Illustration Review 30:15

## Section 5: Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 38:11
Intro 0:00
Eigenvalues and Eigenvectors 0:38
Eigenvalues and Eigenvectors 0:39
Definition 1 3:30
Example 1 7:20
Example 2 10:19
Definition 2 21:15
Example 3 23:41
Theorem 1 26:32
Theorem 2 27:56
Example 4 29:14
Review 34:32
Similar Matrices & Diagonalization 29:55
Intro 0:00
Similar Matrices and Diagonalization 0:25
Definition 1 0:26
Example 1 2:00
Properties 3:38
Definition 2 4:57
Theorem 1 6:12
Example 3 9:37
Theorem 2 12:40
Example 4 19:12
Example 5 20:55
Procedure for Diagonalizing Matrix A: Step 1 24:21
Procedure for Diagonalizing Matrix A: Step 2 25:04
Procedure for Diagonalizing Matrix A: Step 3 25:38
Procedure for Diagonalizing Matrix A: Step 4 27:02
Diagonalization of Symmetric Matrices 30:14
Intro 0:00
Diagonalization of Symmetric Matrices 1:15
Diagonalization of Symmetric Matrices 1:16
Theorem 1 2:24
Theorem 2 3:27
Example 1 4:47
Definition 1 6:44
Example 2 8:15
Theorem 3 10:28
Theorem 4 12:31
Example 3 18:00

## Section 6: Linear Transformations

Linear Mappings Revisited 24:05
Intro 0:00
Linear Mappings 2:08
Definition 2:09
Linear Operator 7:36
Projection 8:48
Dilation 9:40
Contraction 10:07
Reflection 10:26
Rotation 11:06
Example 1 13:00
Theorem 1 18:16
Theorem 2 19:20
Kernel and Range of a Linear Map, Part I 26:38
Intro 0:00
Kernel and Range of a Linear Map 0:28
Definition 1 0:29
Example 1 4:36
Example 2 8:12
Definition 2 10:34
Example 3 13:34
Theorem 1 16:01
Theorem 2 18:26
Definition 3 21:11
Theorem 3 24:28
Kernel and Range of a Linear Map, Part II 25:54
Intro 0:00
Kernel and Range of a Linear Map 1:39
Theorem 1 1:40
Example 1: Part A 2:32
Example 1: Part B 8:12
Example 1: Part C 13:11
Example 1: Part D 14:55
Theorem 2 16:50
Theorem 3 23:00
Matrix of a Linear Map 33:21
Intro 0:00
Matrix of a Linear Map 0:11
Theorem 1 1:24
Procedure for Computing to Matrix: Step 1 7:10
Procedure for Computing to Matrix: Step 2 8:58
Procedure for Computing to Matrix: Step 3 9:50
Matrix of a Linear Map: Property 10:41
Example 1 14:07
Example 2 18:12
Example 3 24:31

Duration: 23 hours, 32 minutes

Number of Lessons: 34

This course is essential for college students taking Linear Algebra who want to learn both theory and application. Each lesson in the course begins with essential ideas & proofs before concluding with many worked-out examples. Theorems are stated and used but not proved to keep the emphasis on problem solving and gaining an intuitive feeling for the concepts.

• Free Sample Lessons
• Closed Captioning (CC)

Topics Include:

• Dot Products
• Determinants
• Linear Transformations
• Subspaces
• Spanning Set for a Vector Space
• Rank of a Matrix
• Orthogonal Complements
• Eigenvalues & Eigenvectors
• Linear Mappings
• Kernel & Range

Professor Hovasapian combines his triple degrees in Mathematics (University of Utah), Chemistry (UC Irvine), and Classics (UC Irvine), with 15+ years of teaching and tutoring experience to help students understand difficult mathematical concepts in Linear Algebra.

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#### Student Feedback

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53 Reviews

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By Levon GuyumjanOctober 11, 2017
I truly enjoy your explanation, is detailed and very clear. Thank so much
By John LinsJanuary 31, 2017
You are the best!! Thank you. I am emailing you now.
By Growth Mindset BelieverApril 3, 2016
Thanks for the great response.  I let my subscription lapse which is why I haven't seen your response until now, but I'm back to enjoy your videos and others on this site again since I'm taking a data science course right now which uses a lot of concepts from linear algebra.
By David LÃ¶fqvistMarch 19, 2016
Maybe wrong place again, but could you explain the vector triple product? I know hos to calculate it, but I havet no idea och what I'm calculating?
Thanks again for your great lessons! I was writing on my tablet and the autocorrect function isn't that good...
By David LÃ¶fqvistMarch 19, 2016
Maybe wrong place again, but could you explain the Hector triple production? I know hos to calcu?ate it, but I havet no idea och what I'm calculating?
Thanks again for your great Messina!

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