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

DIGITAL AUDIOVISUAL LECTURES
 

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 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, 0836916, 0836805, 1322586.  Any opinions, findings, and conclusions orCreative Commons License 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.