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 RowEquivalent 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 NonSingular 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 NonSingular Equivalences 
28:37  
 
 Example: Does the Following System Have a Nontrivial 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 15 
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 NonSingular Equivalences 
43:07  
 
 List of NonSingular 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  

nVector 
52:44 
 
Intro 
0:00  
 
nVectors 
0:58  
 
 4Vector 
0:59  
 
 7Vector 
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  
 
 CauchySchwarz Inequality 
24:56  
 
 Triangle Inequality 
36:29  
 
 Unit Vector 
40:34  
 
Vectors and Dot Products 
44:23  
 
 Orthogonal Vectors 
44:24  
 
 CauchySchwarz 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 NonHomogeneous 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 NonSingular Equivalences 
24:24  
 
 List of NonSingular 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 nSpace 
32:53 
 
Intro 
0:00  
 
Orthonormal Bases in nSpace 
1:02  
 
 Orthonormal Bases in nSpace: 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  