Professor Raffi Hovasapian covers Educator's Linear Algebra course with topics spanning everything from Matrices to Eigenvectors. Each lesson begins with essential theory and is anchored with many worked-out examples. Other topics in this fully self-contained Linear Algebra course include Determinants, Linear Transformations, Homogeneous Systems, and Orthogonal Complements. Professor Hovasapian combines his triple degrees in Mathematics, Chemistry, and Classics with his 10+ years of teaching experience to help students understand difficult mathematical concepts.

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I. 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 28:54
   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 
II. 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 
III. 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 
IV. 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 
V. 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 
VI. 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