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Linear Algebra Online Course Prof. Raffi Hovasapian
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34 Lessons (23hr : 32min)
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5.0 629 Ratings & 53 Reviews
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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.

Table of Contents
Course Details
About Prof. Hovasapian

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
			Matrix Addition	10:36
			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 Addition		1:11
			Properties of Addition: A	1:12
			Properties of Addition: B	2:30
			Properties of Addition: C	2:57
			Properties of Addition: D	4:20
		Properties of Addition		5:22
			Properties of Addition	5:23
		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
		Properties of Matrix Addition		33:25
			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
			Vector Addition	19:34
			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
			Vector Addition	2:43
			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.

Additional Features:

Free Sample Lessons
Closed Captioning (CC)
Downloadable Lecture Slides
Instructor Comments

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.

Student Testimonials:

"My professor should watch these lectures so he can learn how to teach!" — Brodey H.

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"Dr. Hovasapian actually has the ability to organize a course, he knows how to present material in an educational manner, and in the one video I learned more than I did in several weeks of my university lectures." — Jason M.

“Correct me if I'm wrong but I can tell you have been sitting and thinking deeply about all these complicated expressions from whatever textbook you have until you got a breakthrough and really grasped it. That's why you are so amazingly good at this, and I really appreciate it!! Best regards!” — Christian F.

“I really can’t express how useful your lectures were to me, I really wouldn't do well in my class if I never found your lectures on this site. It's so worth it. Thank you so much Professor Hovasapian!” — Hengyao H.

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Student Feedback

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53 Reviews

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By Levon Guyumjan•October 11, 2017

I truly enjoy your explanation, is detailed and very clear. Thank so much

By John Lins•January 31, 2017

You are the best!! Thank you. I am emailing you now.

By Growth Mindset Believer•April 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¶fqvist•March 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¶fqvist•March 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!