# Introduction to Matrix Algebra

A just-in-time tool for various STEM courses and a much needed refresher!
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68 students
Introduction to Matrix Algebra
Rating: 0.0 out of 5 (0 ratings)
68 students
know vectors and their linear combinations and dot products
know why we need matrix algebra and differentiate between various special matrices
carry unary operations on matrices
carry binary operations on matrices
differentiate between inconsistent and consistent system of equations via finding rank of matrices
differentiate between unique and infinite solution system of equations
use Gaussian elimination methods to find solution to a system of equations
use LU decomposition to find solution to system of equations and know when to choose the method over Gaussain elimination
use Gauss-Seidel method to solve a system of equations iteratively
find quantitatively how adequate your solution is through the concept of condition numbers
find eigenvectors and eigenvalues of a square matrix

### Requirements

• College Algebra
Description

Matrix algebra is used in a very diverse field of studies. Some of these fields include engineering, mathematics, and business. This course starts with the basics of matrix algebra with questions like: "What is a vector?" No precursory knowledge about matrix algebra is required on the part of the student, so not to worry if you are new to the subject! If you already have some knowledge of beginner concepts, just skip to the area of the course that's right for you! The video lectures are short; covering only one topic at a time, so it's easy to jump right to your level of knowledge.

The course has several important components that are all essential to the student's understanding of the material.

Textbook: Each section or chapter will start with the textbook chapter for that section.

Video Lectures: Next, there will be a series of video lectures; one micro lecture per topic. There are several types of video lectures, the two most common being theory or example (usually in that order). First, Dr. Kaw will talk about the theory or background behind a particular concept or topic. He will then proceed to work out an example using that concept.

Practice Problems: Each section will be concluded with a set of practice problems. These practice problems are meant to give the student a medium of testing their mastery of the concepts. Combined with these practice problems are the full solutions to each question. These solutions can be used to check your approach and final answer.

Who this course is for:
• Students who are in a STEM major in college. It is also suited for finance and economics majors. If your exposure to college algebra is limited, this course is not for you!
Course content
10 sections177 lectures14h 56m total length
• Textbook Chapter 1
8 pages
• Definition of a Matrix
02:21
• Definition of a Square Matrix
01:43
• Definition of a Submatrix
02:43
• Diagonal Matrix
02:45
• Diagonally Dominant Matrix
07:21
• Identity Matrix
02:01
• Lower Triangular Matrix
03:13
• Equal Matrices
02:53
• Column Vector
01:23
• Row Vector
01:23
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
8 pages
• Textbook Chapter 2
16 pages
• Definition of a Vector: Theory
02:26
• Definition of a Vector: Example
01:14
01:33
01:47
• Multiply a Vector By a Scalar: Theory
01:17
• Multiply a Vector By a Scalar: Example
01:15
• Dot Product: Theory
01:59
• Dot Product: Example
01:40
• Rank of a Set of Vectors: Theory
02:08
• Rank of a Set of Vectors: Example 1
03:52
• Rank of a Set of Vectors: Example 2
02:26
• Linear Combination of Vectors: Theory
01:43
• Linear Combination of Vectors: Example
02:33
• Simultaneous Linear Equations in Vector Form: Theory
05:07
• Simultaneous Linear Equations in Vector Form: Example
03:12
• Null or Zero Vectors: Theory
01:11
• Unit Vectors: Theory
01:29
• Unit Vectors: Example
01:48
• Equivalent Vectors: Theory
02:01
• Equivalent Vectors: Example
02:25
• Linearly Dependent Vectors: Proof 1
02:28
• Linearly Dependent Vectors: Proof 2
04:06
• Linearly Independent Vectors: Theory
02:27
• Linearly Independent Vectors: Example 1
03:16
• Linearly Independent Vectors: Example 2
09:12
• Subsets of Linearly Independent Vectors: Proof
05:41
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
7 pages
• Textbook Chapter 3
9 pages
01:53
02:10
• Matrix Subtraction: Theory
01:39
• Matrix Subtraction: Example
02:04
• Linear Combination of Matrices: Theory
02:03
• Linear Combination of Matrices: Example
03:56
• Matrix Multiplication: Theory
04:32
• Matrix Multiplication: Example
06:19
• Is Matrix Multiplication Commutative?
04:00
• Product of a Scalar and a Matrix: Theory
01:36
• Product of a Scalar and a Matrix: Example
01:44
• Rules of Binary Matrix Operations: Part 1
01:46
• Rules of Binary Matrix Operations: Part 2
01:38
• Rules of Binary Matrix Operations: Part 3
02:49
• Rules of Binary Matrix Operations: Part 4
02:30
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
6 pages
• Textbook Chapter 4
10 pages
• Determinant of a Matrix Using Cofactors: Theory
03:25
• Determinant of a Matrix Using Cofactors: Example
05:30
• Determinant of a Matrix Using Minors: Theory
04:40
• Determinant of a Matrix Using Minors: Example
06:25
• Skew-symmetric Matrix
03:11
• Symmetric Matrix
03:19
• Theorems on Determinants: Part 1
01:35
• Theorems on Determinants: Part 2
03:34
• Theorems on Determinants: Part 3
03:20
• Theorems on Determinants: Part 4
02:41
• Trace of a Matrix
02:02
• Transpose of a Matrix
04:11
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
6 pages
• Textbook Chapter 5
18 pages
• Writing Simultaneous Linear Equations in Matrix Form
05:25
• Setting Up Simultaneous Linear Equations: Example
05:22
• Number of Solutions for a System of Linear Equations
04:56
• Consistent and Inconsistent System of Equations: Theory
02:55
• Consistent and Inconsistent System of Equations: Example
05:30
• Consistent and Inconsistent System of Equations
03:05
• Consistent and Inconsistent System of Equations: Example 1
04:14
• Consistent and Inconsistent System of Equations: Example 2
08:41
• Determining the Uniqueness of a Solution
03:24
• Consistent and Inconsistent System of Equations: Example 3
06:06
• Does a Set of Equations Have a Unique Solution: Example 1
02:17
• Does a Set of Equations Have a Unique Solution: Example 2
02:22
• Matrix Division
05:38
• Finding the Inverse of a Matrix: Theory
04:31
• Finding the Inverse of a Matrix: Example
07:02
• Finding the Inverse of a Matrix by Adjoints: Theory
02:14
• Finding the Inverse of a Matrix by Adjoints: Example
07:18
• Uniqueness of a Matrix
02:29
• Does more than one unknown mean inconsistent equations?
08:10
• Inverse of Matrices: Example
03:39
• Rank of a Matrix: Theory
01:20
• Rank of a Matrix: Example 1
01:30
• Rank of a Matrix: Example 2
02:49
• Facts About the Inverse of a Matrix
02:52
• Solving a Set of Equations With the Inverse
02:26
• End of Chapter Practice Problems
3 pages
• Solutions to Practice Problems
10 pages
• Textbook Chapter 6
18 pages
• Naive GE: Theory Part 1
10:27
• Naive GE: Theory Part 2
02:22
• Naive GE: Example 1 Part 1 Forward Elimination
10:49
• Naive GE: Example 2 Part 1 Backward Substitution
08:07
• Naive GE: Example 2 Part 2 Backward Substitution
06:40
• Naive GE: Pitfalls
07:20
• Naive GE: Example of Round Off Error Part 1
07:20
• Naive GE: Example of Round Off Error Part 2
07:40
• GE With Partial Pivoting: Theory
10:39
• GE With Partial Pivoting: Example Part 1
07:15
• GE With Partial Pivoting: Example Part 2
10:07
• GE With Partial Pivoting: Example Part 3
06:17
• GE w/ Partial Pivoting: Example of Round Off Error Part 1
08:58
• GE w/ Partial Pivoting: Example of Round Off Error Part 2
08:17
• GE w/ Partial Pivoting: Example of Round Off Error Part 3
05:47
• Determinant of a Matrix Using FE: Background
05:17
• Determinant of a Matrix Using FE: Example
10:07
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
9 pages
• Textbook Chapter 7
10 pages
• LU Decomposition Basis
09:02
• Finding the Inverse of a Matrix: Theory
06:02
• Finding the Inverse of a Matrix: Example
10:20
• LU Decoposition: Example Part 1
06:55
• LU Decoposition: Example Part 1
04:36
• Solving a Set of Equations: Example
10:28
• Advantages of LU Decomposition Part 1
04:57
• Advantages of LU Decomposition Part 2
08:05
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
11 pages
• Textbook Chapter 8
10 pages
• Gauss-Seidel Method: Theory Part 1
08:00
• Gauss-Seidel Method: Theory Part 2
05:37
• Gauss-Seidel Method: Example Part 1
09:16
• Gauss-Seidel Method: Example Part 2
07:39
• Gauss-Seidel Method: Pitfall Part 1
07:50
• Gauss-Seidel Method: Pitfall Part 2
08:13
• End of Chapter Practice Problems
1 page
• Solutions to Practice Problems
11 pages
• Textbook Chapter 9
11 pages
• Properties of Norms
03:36
• Relation of Norm to the Conditioning of SLEs: Part 1
08:54
• Relation of Norm to the Conditioning of SLEs: Part 2
05:57
• Ill-conditioned and Well-conditioned SLEs
10:11
• Significant Digits in Solution Vector: Theory
03:59
• Significant Digits In Solution Vector: Example 1
03:56
• Significant Digits In Solution Vector: Example 2
04:25
• Relate Changes in Coef Matrix to Changes in Soln: Proof
08:45
• Relating Changes in Coeff Matrix to Changes in Soln Vec
04:17
• Relating Changes in RHS Vec to Changes in Solution Vec
03:11
• Row Sum Norm of a Matrix: Example
03:05
• Row Sum Norm of a Matrix: Theory Test 2
02:33
• Row Sum Norm of a Matrix: Theory
02:33
• End of Chapter Practice Problems
3 pages
• Solutions to Practice Problems
9 pages
• Textbook Chapter 10
12 pages
• Origin of the Word Eigenvalue
01:01
• Theorems of Eigenvalues and Eigenvectors: Part 1
02:18
• Theorems of Eigenvalues and Eigenvectors: Part 2
02:05
• Theorems of Eigenvalues and Eigenvectors: Part 3
02:43
• Theorems of Eigenvalues and Eigenvectors: Part 4
00:52
• Theorems of Eigenvalues and Eigenvectors: Part 5
01:36
• Theorems of Eigenvalues and Eigenvectors: Part 6
03:14
• Definition of Eigenvalues and Eigenvectors
03:10
• Eigenvalues of a Square Matrix: Theory
04:32
• Eigenvalues of a Square Matrix: Example
03:45
• Eigenvectors of a Square Matrix: Example
06:32
• Eigenvectors of a Square Matrix: Example 2
13:09
• Find Eigenvalues and Eigenvectors Numerically: Theory
04:56
• Find Eigenvalues and Eigenvectors Numerically: Example
08:08
• Application of Eigenvalues and Eigenvectors
16:22
• End of Chapter Practice Problems
2 pages
• Solutions to Practice Problems
7 pages

Instructor
A Global Teacher

Autar Kaw is a professor of mechanical engineering and Jerome Krivanek Distinguished Teacher at the University of South Florida. He is a recipient of the 2012 U.S. Professor of the Year Award from the Council for Advancement and Support of Education (CASE) and Carnegie Foundation for Advancement of Teaching.

Professor Kaw received his BE Honors degree in Mechanical Engineering from Birla Institute of Technology and Science (BITS) India in 1981, and his degrees of Ph.D. in 1987 and M.S. in 1984, both in Engineering Mechanics from Clemson University, SC. He joined University of South Florida in 1987.

Professor Kaw’s main scholarly interests are in engineering education research, open courseware development, bascule bridge design, fracture mechanics, composite materials, computational nanomechanics, and the state and future of higher education. His research has been funded by National Science Foundation, Air Force Office of Scientific Research, Florida Department of Transportation, Research and Development Laboratories, Wright Patterson Air Force Base, and Montgomery Tank Lines.

Professor Kaw has written several books on subjects such as composite materials, numerical methods, computer programming, matrix algebra, and engineering licensure examination.

Since 2002, under Professor Kaw's leadership, he and his colleagues from around the nation have developed, implemented, refined and assessed online resources for open courseware in Numerical Methods. This courseware annually receives more than a million page views, 900,000 views of the YouTube lectures and 150,000 annual visitors to the "numerical methods guy" blog.

Professor Kaw's work has appeared in the St. Petersburg Times, Tampa Tribune, Chance, Oracle, and his work has been covered/cited in Chronicle of Higher Education, Inside Higher Education, Congressional Record, ASEE Prism, Tampa Bay Times, Tampa Tribune, Campus Technology, Florida Trend Magazine, WUSF, Bay News 9, Times of India, NSF Discoveries, Voice of America, and Indian Express.

Professor Kaw is a Fellow of the American Society of Mechanical Engineers (ASME) and a member of the American Society of Engineering Education (ASEE). He has also been a Maintenance Engineer (1982) for Ford-Escorts Tractors, India, and a Summer Faculty Fellow (1992) and Visiting Scientist (1991) at Wright Patterson Air Force Base.