Digital Audiovisual Lectures – Introduction to Matrix Algebra

Chapter 01: Introduction [MORE]

- - What is a matrix? [YOUTUBE 2:22] [TRANSCRIPT]

- - What is a row vector? [YOUTUBE 1:24] [TRANSCRIPT]
  - What is a column vector? [YOUTUBE 1:58] [TRANSCRIPT]

- - What is a submatrix? [YOUTUBE 2:44] [TRANSCRIPT]

- - What is a square matrix? [YOUTUBE 1:44] [TRANSCRIPT]

- - What is an upper triangular matrix? [YOUTUBE 3:02] [TRANSCRIPT]
  - What is a lower triangular matrix? [YOUTUBE 3:14] [TRANSCRIPT]
  - What is an identity matrix? [YOUTUBE 2:02] [TRANSCRIPT]
  - Diagonally dominant matrix [YOUTUBE 7:22] [TRANSCRIPT]

- - Equal matrices [YOUTUBE 2:54] [TRANSCRIPT]

Chapter 02: Vectors [MORE]

- - What is a vector? [YOUTUBE 2:27] [TRANSCRIPT]
  - What is a vector? Example [YOUTUBE 1:15] [TRANSCRIPT]

- - When are two vectors equal? [YOUTUBE 2:02][TRANSCRIPT]
  - When are two vectors equal? Example [YOUTUBE 2:26][TRANSCRIPT]

- - How do you add two vectors? [YOUTUBE 1:34][TRANSCRIPT]
  - How do you add two vectors? Example [YOUTUBE 1:48] [TRANSCRIPT]

- - What is a null (or zero) vector? [YOUTUBE 1:12] [TRANSCRIPT]
  - What is a unit vector? [YOUTUBE 1:30][TRANSCRIPT]
  - What is a unit vector? Example [YOUTUBE 1:49] [TRANSCRIPT]

- - How do you multiply a vector by a scalar? [YOUTUBE 1:18] [TRANSCRIPT]
  - How do you multiply a vector by a scalar? Example [YOUTUBE 1:16] [TRANSCRIPT]

- - What do you mean by linear combination of vectors? [YOUTUBE 1:44] [TRANSCRIPT]
  - What do you mean by linear combination of vectors? Example [YOUTUBE 2:34][TRANSCRIPT]
  - What do you mean by vectors being linearly independent? [YOUTUBE 2:28] [TRANSCRIPT]
  - Are these vectors linearly independent? Example 1 [YOUTUBE 3:17] [TRANSCRIPT]
  - Are these vectors linearly independent? Example 2 [YOUTUBE 9:13][TRANSCRIPT]

- - What do you mean by rank of a set of vectors? [YOUTUBE 2:09] [TRANSCRIPT]
  - Rank of a set of vectors: Example 1 [YOUTUBE 3:53] [TRANSCRIPT]
  - Rank of a set of vectors: Example 2 [YOUTUBE 2:27] [TRANSCRIPT]

- - Prove that if a set of vectors contains a null vector, the set of vectors is linearly dependent [YOUTUBE 2:29] [TRANSCRIPT]
  - Prove that if a set of vectors is linearly independent, then a subset of it is also linearly independent [YOUTUBE 5:42][TRANSCRIPT]
  - Prove that if a set of vectors is linearly dependent, then at least one vector can be written as a linear combination of others [YOUTUBE 4:06] [TRANSCRIPT]

- - How can vectors be used to write simultaneous linear equations? [YOUTUBE 5:08] [TRANSCRIPT]
  - How can vectors be used to write simultaneous linear equations? Example [YOUTUBE 3:13][TRANSCRIPT]

- - What is the definition of the dot product of two vectors? [YOUTUBE 2:00] [TRANSCRIPT]
  - Dot product of two vectors Example [YOUTUBE 1:41] [TRANSCRIPT]

Chapter 03: Binary Matrix Operations [MORE]

- - Adding two matrices Theory [YOUTUBE 1:54] [TRANSCRIPT]
  - Adding two matrices Example [YOUTUBE 2:11] [TRANSCRIPT]

- - Subtracting two matrices Theory [YOUTUBE 1:40][TRANSCRIPT]
  - Subtracting two matrices Example [YOUTUBE 2:05][TRANSCRIPT]

- - Multiplying two matrices Theory [YOUTUBE 4:33][TRANSCRIPT]
  - Multiplying two matrices Example [YOUTUBE 6:20] [TRANSCRIPT]

- - Product of a scalar and a matrix Theory [YOUTUBE 1:37][TRANSCRIPT]
  - Product of a scalar and a matrix Example [YOUTUBE 1:45] [TRANSCRIPT]

- - Linear combination of matrices Theory [YOUTUBE 2:04] [TRANSCRIPT]
  - Linear combination of matrices Example [YOUTUBE 3:57][TRANSCRIPT]

- - Rules of binary matrix operations Part 1 of 4 [YOUTUBE 1:47] [TRANSCRIPT]
  - Rules of binary matrix operations Part 2 of 4 [YOUTUBE 1:38][TRANSCRIPT]
  - Rules of binary matrix operations Part 3 of 4 [YOUTUBE 2:50] [TRANSCRIPT]
  - Rules of binary matrix operations Part 4 of 4 [YOUTUBE 2:31] [TRANSCRIPT]
  - Is matrix multiplication commutative? [YOUTUBE 4:01][TRANSCRIPT]

Chapter 04: Unary Matrix Operations [MORE]

- - Transpose of a matrix [YOUTUBE 4:12] [TRANSCRIPT]
  - Symmetric matrix [YOUTUBE 3:20] [TRANSCRIPT]
  - Skew symmetric matrix [YOUTUBE 3:12][TRANSCRIPT]

- - Trace of a matrix [YOUTUBE 2:03][TRANSCRIPT]

- - Determinant of a matrix using minors Theory [YOUTUBE 4:41][TRANSCRIPT]
  - Determinant of a matrix using cofactors Theory [YOUTUBE 3:26] [TRANSCRIPT]
  - Determinant of a matrix using minors Example [YOUTUBE 6:26][TRANSCRIPT]
  - Determinant of a matrix using cofactors Example [YOUTUBE 5:31] [TRANSCRIPT]
  - Theorems on determinants Part 1 of 4 [YOUTUBE 1:36] [TRANSCRIPT]
  - Theorems on determinants Part 2 of 4 [YOUTUBE 3:35] [TRANSCRIPT]
  - Theorems on determinants Part 3 of 4 [YOUTUBE 3:21] [TRANSCRIPT]
  - Theorems on determinants Part 4 of 4 [YOUTUBE 2:42][TRANSCRIPT]

Chapter 05: System of Equations [MORE]

- - A real life problem of setting up simultaneous linear equations [YOUTUBE 5:23] [TRANSCRIPT]
  - Writing simultaneous linear equations in matrix form [YOUTUBE 5:25] [TRANSCRIPT]

- - Consistent and inconsistent system of equations Theory [YOUTUBE 2:56] [TRANSCRIPT]
  - Consistent and inconsistent system of equations Example [YOUTUBE 5:31] [TRANSCRIPT]

- - Rank of a matrix Definition [YOUTUBE 1:21] [TRANSCRIPT]
  - Rank of a matrix Example 1 [YOUTUBE 1:31] [TRANSCRIPT]
  - Rank of a matrix Example 2 [YOUTUBE 2:50] [TRANSCRIPT]

- - Distinguishing between consistent and inconsistent system of equations based on rank of matrices [YOUTUBE 3:06] [TRANSCRIPT]
  - Distinguishing between consistent and inconsistent system of equations based on rank of matrices Example 1 [YOUTUBE 4:15] [TRANSCRIPT]
  - Distinguishing between consistent and inconsistent system of equations based on rank of matrices Example 2 [YOUTUBE 8:42] [TRANSCRIPT]
  - Distinguishing between consistent and inconsistent system of equations based on rank of matrices Example 3 [YOUTUBE 6:07] [TRANSCRIPT]

- - If a solution exists, how do we know if it is unique? [YOUTUBE 3:25] [TRANSCRIPT]
  - Does a set of equations have a unique solution? Example 1 [YOUTUBE 2:18] [TRANSCRIPT]
  - Does a set of equations have a unique solution? Example 2 [YOUTUBE 2:23] [TRANSCRIPT]
  - If we have more equations than unknowns, does it mean we have inconsistent system of equations? [YOUTUBE 8:11] [TRANSCRIPT]
  - Can a system of equations have more then one but not infinite number of solutions? [YOUTUBE 4:57] [TRANSCRIPT]

- - Can we divide two matrices? [YOUTUBE 5:39] [TRANSCRIPT]
  - Some statements about the inverse of matrices [YOUTUBE 2:53] [TRANSCRIPT]
  - Inverse of matrices Example [YOUTUBE 3:40] [TRANSCRIPT]
  - Using concept of inverse to solve a set of equations [YOUTUBE 2:27] [TRANSCRIPT]
  - Finding the inverse of a matrix Theory [YOUTUBE 4:32] [TRANSCRIPT]
  - Finding the inverse of a matrix Example [YOUTUBE 7:03] [TRANSCRIPT]
  - Finding the inverse of a matrix by adjoints Theory [YOUTUBE 2:15] [TRANSCRIPT]
  - Finding the inverse of a matrix by adjoints Example [YOUTUBE 7:19] [TRANSCRIPT]
  - If the inverse of [A] exists, is it unique? [YOUTUBE 2:30] [TRANSCRIPT]

Chapter 06: Gaussian Elimination [MORE]

- - Naive Gaussian elimination: Theory: Part 1 of 2 [YOUTUBE 10:27] [TRANSCRIPT]
  - Naive Gaussian elimination: Theory: Part 2 of 2 [YOUTUBE 2:22] [TRANSCRIPT]
  - Naive Gauss Elimination Method: Example: Part 1 of 2 (Forward Elimination) [YOUTUBE 10:49] [TRANSCRIPT]
  - Naive Gauss Elimination Method: Example: Part 2 of 2 (Back Substitution) [YOUTUBE 6:40] [TRANSCRIPT]

- - Pitfalls of Naive Gauss Elimination Method: [YOUTUBE 7:20] [TRANSCRIPT]
  - Naive Gauss Elimination: Round-off Error Issues: Example: Part 1 of 3 [YOUTUBE 7:20] [TRANSCRIPT]
  - Naive Gauss Elimination: Round-off Error Issues: Example: Part 2 of 3 [YOUTUBE 7:40] [TRANSCRIPT]
  - Naive Gauss Elimination: Round-off Error Issues: Example: Part 3 of 3 [YOUTUBE 8:07] [TRANSCRIPT]

- - Gaussian Elimination With Partial Pivoting: Theory [YOUTUBE 10:39] [TRANSCRIPT]
  - Gaussian Elimination With Partial Pivoting: Example: Part 1 of 3 (Forward Elimination) [YOUTUBE 7:15] [TRANSCRIPT]
  - Gaussian Elimination With Partial Pivoting: Example: Part 2 of 3 (Forward Elimination) [YOUTUBE 10:08] [TRANSCRIPT]
  - Gaussian Elimination With Partial Pivoting: Example: Part 3 of 3 (Back Substitution) [YOUTUBE 6:18] [TRANSCRIPT]

- - Gaussian Elimination With Partial Pivoting: Round-off Error Issues: Example: Part 1 of 3 [YOUTUBE 8:58] [TRANSCRIPT]
  - Gaussian Elimination With Partial Pivoting: Round-off Error Issues: Example: Part 2 of 3 [YOUTUBE 8:17] [TRANSCRIPT]
  - Gaussian Elimination With Partial Pivoting: Round-off Error Issues: Example: Part 3 of 3 [YOUTUBE 5:48] [TRANSCRIPT]

- - Determinant of a Matrix Using Forward Elimination Method: Background [YOUTUBE 5:17] [TRANSCRIPT]
  - Determinant of a Matrix Using Forward Elimination Method: Example [YOUTUBE 10:07] [TRANSCRIPT]

Chapter 07: LU Decomposition [MORE]

- - LU Decomposition: Basis [YOUTUBE 9:02] [TRANSCRIPT]
  - LU Decomposition Method: Example [YOUTUBE 10:29] [TRANSCRIPT]
  - Why LU Decomposition: Part 1 [YOUTUBE 4:58] [TRANSCRIPT]
  - Why LU Decomposition: Part 2 [YOUTUBE 8:05] [TRANSCRIPT]
  - Decomposing a Square Matrix: Part 1 of 2 [YOUTUBE 6:56] [TRANSCRIPT]
  - Decomposing a Square Matrix: Part 2 of 2 [YOUTUBE 4:37] [TRANSCRIPT]
  - Finding Inverse of a Matrix Using LU Decomposition: Background [YOUTUBE 6:03] [TRANSCRIPT]
  - Finding Inverse of a Matrix Using LU Decomposition: Example [YOUTUBE 10:20] [TRANSCRIPT]

Chapter 08: Gauss-Seidel Method [MORE]

- - Gauss-Seidel Method of Solving Simul Linear Eqns: Theory: Part 1 of 2 [YOUTUBE 8:01] [TRANSCRIPT]
  - Gauss-Seidel Method of Solving Simul Linear Eqns: Theory: Part 2 of 2 [YOUTUBE 5:38] [TRANSCRIPT]
  - Gauss-Seidel Method of Solving Simul Linear Eqns: Example: Part 1 of 2 [YOUTUBE 9:17] [TRANSCRIPT]
  - Gauss-Seidel Method of Solving Simul Linear Eqns: Example: Part 2 of 2 [YOUTUBE 7:40] [TRANSCRIPT]
  - Gauss-Seidel Method of Solving Simul Linear Eqns:
    - - Pitfalls and Advantages: Part 1 of 2 [YOUTUBE 7:50] [TRANSCRIPT]
  - Gauss-Seidel Method of Solving Simul Linear Eqns:
    - - Pitfalls and Advantages: Part 2 of 2 [YOUTUBE 8:14] [TRANSCRIPT]

Chapter 09: Adequacy of Solutions [MORE]

- - ill-conditioned and well-conditioned system of equations [YOUTUBE 10:12] [TRANSCRIPT]

- - Row sum norm of a matrix Theory [YOUTUBE 2:34] [TRANSCRIPT]
  - Row sum norm of a matrix Example [YOUTUBE 3:06] [TRANSCRIPT]
  - How is the norm related to the conditioning of a system of equations Part 1 of 2 [YOUTUBE 8:55] [TRANSCRIPT]
  - How is the norm related to the conditioning of a system of equations Part 2 of 2 [YOUTUBE 5:58] [TRANSCRIPT]
  - Properties of norms [YOUTUBE 3:37] [TRANSCRIPT]

- - Relating changes in coefficient matrix to changes in solution vector [YOUTUBE 4:18] [TRANSCRIPT]
  - Relating changes in coefficient matrix to changes in solution vector Proof [YOUTUBE 8:46] [TRANSCRIPT]
  - Relating changes in right hand side vector to changes in solution vector [YOUTUBE 3:12] [TRANSCRIPT]

- - Number of significant digits correct in my solution vector Theory [YOUTUBE 4:00] [TRANSCRIPT]
  - Number of significant digits correct in my solution vector Example 1 [YOUTUBE 3:57] [TRANSCRIPT]
  - Number of significant digits correct in my solution vector Example 2 [YOUTUBE 4:26] [TRANSCRIPT]

Chapter 10: Eigenvalues and Eigenvectors [MORE]

- - What is the origin of the word eigenvalue [YOUTUBE 1:02] [TRANSCRIPT]

- - A physical example of application of eigenvalues and eigenvectors [YOUTUBE 16:23] [TRANSCRIPT]

- - Definition of eigenvalues and eigenvectors [YOUTUBE 3:11] [TRANSCRIPT]

- - How do I find eigenvalues of a square matrix [YOUTUBE 4:33][TRANSCRIPT]
  - How do I find eigenvalues of a square matrix? Example [YOUTUBE 3:46][TRANSCRIPT]
  - How do I find eigenvectors of a square matrix? Example [YOUTUBE 6:33][TRANSCRIPT]
  - How do I find eigenvectors of a square matrix? Example 2 [YOUTUBE 13:10] [TRANSCRIPT]

- - Theorems of eigenvalues and eigenvectors Part 1 of 6 [YOUTUBE 2:19] [TRANSCRIPT]
  - Theorems of eigenvalues and eigenvectors Part 2 of 6 [YOUTUBE 2:06][TRANSCRIPT]
  - Theorems of eigenvalues and eigenvectors Part 3 of 6 [YOUTUBE 2:43] [TRANSCRIPT]
  - Theorems of eigenvalues and eigenvectors Part 4 of 6 [YOUTUBE 0:53] [TRANSCRIPT]
  - Theorems of eigenvalues and eigenvectors Part 5 of 6 [YOUTUBE 1:37] [TRANSCRIPT]
  - Theorems of eigenvalues and eigenvectors Part 6 of 6 [YOUTUBE 3:15] [TRANSCRIPT]

- - How does one find eigenvalues and eigenvectors numerically [YOUTUBE 4:57] [TRANSCRIPT]
  - How does one find eigenvalues and eigenvectors numerically Example [YOUTUBE 8:09][TRANSCRIPT]