INTRODUCTION TO MATRIX ALGEBRA Transforming Matrix Algebra for the STEM Undergraduate HOME | TOPICS | VIDEOS | TEXTBOOK | PPT | QUIZZES | PROBLEM SET | CONTACT | MATH FOR COLLEGE | NUMERICAL METHODS
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 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 03.   BINARY 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 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]

 Copyrights: University of South Florida, 4202 E Fowler Ave, Tampa, FL 33620-5350. All Rights Reserved. Questions, suggestions or comments, contact kaw@usf.edu  This material is based upon work supported by the National Science Foundation under Grant# 0126793, 0341468, 0717624,  0836981, , 0836805, 1322586.  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.  Based on a work at http://mathforcollege.com/ma.  Introduction to Matrix Algebra is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.