# Introduction to Matrix Algebra

### Requirements

- College Algebra

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.

- 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!

- Textbook Chapter 18 pages

- Definition of a Matrix02:21

- Definition of a Square Matrix01:43

- Definition of a Submatrix02:43

- Diagonal Matrix02:45

- Diagonally Dominant Matrix07:21

- Identity Matrix02:01

- Lower Triangular Matrix03:13

- Equal Matrices02:53

- Column Vector01:23

- Row Vector01:23

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems8 pages

- Textbook Chapter 216 pages

- Definition of a Vector: Theory02:26

- Definition of a Vector: Example01:14

- Vector Addition: Theory01:33

- Vector Addition: Example01:47

- Multiply a Vector By a Scalar: Theory01:17

- Multiply a Vector By a Scalar: Example01:15

- Dot Product: Theory01:59

- Dot Product: Example01:40

- Rank of a Set of Vectors: Theory02:08

- Rank of a Set of Vectors: Example 103:52

- Rank of a Set of Vectors: Example 202:26

- Linear Combination of Vectors: Theory01:43

- Linear Combination of Vectors: Example02:33

- Simultaneous Linear Equations in Vector Form: Theory05:07

- Simultaneous Linear Equations in Vector Form: Example03:12

- Null or Zero Vectors: Theory01:11

- Unit Vectors: Theory01:29

- Unit Vectors: Example01:48

- Equivalent Vectors: Theory02:01

- Equivalent Vectors: Example02:25

- Linearly Dependent Vectors: Proof 102:28

- Linearly Dependent Vectors: Proof 204:06

- Linearly Independent Vectors: Theory02:27

- Linearly Independent Vectors: Example 103:16

- Linearly Independent Vectors: Example 209:12

- Subsets of Linearly Independent Vectors: Proof05:41

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems7 pages

- Textbook Chapter 39 pages

- Matrix Addition: Theory01:53

- Matrix Addition: Example02:10

- Matrix Subtraction: Theory01:39

- Matrix Subtraction: Example02:04

- Linear Combination of Matrices: Theory02:03

- Linear Combination of Matrices: Example03:56

- Matrix Multiplication: Theory04:32

- Matrix Multiplication: Example06:19

- Is Matrix Multiplication Commutative?04:00

- Product of a Scalar and a Matrix: Theory01:36

- Product of a Scalar and a Matrix: Example01:44

- Rules of Binary Matrix Operations: Part 101:46

- Rules of Binary Matrix Operations: Part 201:38

- Rules of Binary Matrix Operations: Part 302:49

- Rules of Binary Matrix Operations: Part 402:30

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems6 pages

- Textbook Chapter 410 pages

- Determinant of a Matrix Using Cofactors: Theory03:25

- Determinant of a Matrix Using Cofactors: Example05:30

- Determinant of a Matrix Using Minors: Theory04:40

- Determinant of a Matrix Using Minors: Example06:25

- Skew-symmetric Matrix03:11

- Symmetric Matrix03:19

- Theorems on Determinants: Part 101:35

- Theorems on Determinants: Part 203:34

- Theorems on Determinants: Part 303:20

- Theorems on Determinants: Part 402:41

- Trace of a Matrix02:02

- Transpose of a Matrix04:11

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems6 pages

- Textbook Chapter 518 pages

- Writing Simultaneous Linear Equations in Matrix Form05:25

- Setting Up Simultaneous Linear Equations: Example05:22

- Number of Solutions for a System of Linear Equations04:56

- Consistent and Inconsistent System of Equations: Theory02:55

- Consistent and Inconsistent System of Equations: Example05:30

- Consistent and Inconsistent System of Equations03:05

- Consistent and Inconsistent System of Equations: Example 104:14

- Consistent and Inconsistent System of Equations: Example 208:41

- Determining the Uniqueness of a Solution03:24

- Consistent and Inconsistent System of Equations: Example 306:06

- Does a Set of Equations Have a Unique Solution: Example 102:17

- Does a Set of Equations Have a Unique Solution: Example 202:22

- Matrix Division05:38

- Finding the Inverse of a Matrix: Theory04:31

- Finding the Inverse of a Matrix: Example07:02

- Finding the Inverse of a Matrix by Adjoints: Theory02:14

- Finding the Inverse of a Matrix by Adjoints: Example07:18

- Uniqueness of a Matrix02:29

- Does more than one unknown mean inconsistent equations?08:10

- Inverse of Matrices: Example03:39

- Rank of a Matrix: Theory01:20

- Rank of a Matrix: Example 101:30

- Rank of a Matrix: Example 202:49

- Facts About the Inverse of a Matrix02:52

- Solving a Set of Equations With the Inverse02:26

- End of Chapter Practice Problems3 pages

- Solutions to Practice Problems10 pages

- Textbook Chapter 618 pages

- Naive GE: Theory Part 110:27

- Naive GE: Theory Part 202:22

- Naive GE: Example 1 Part 1 Forward Elimination10:49

- Naive GE: Example 2 Part 1 Backward Substitution08:07

- Naive GE: Example 2 Part 2 Backward Substitution06:40

- Naive GE: Pitfalls07:20

- Naive GE: Example of Round Off Error Part 107:20

- Naive GE: Example of Round Off Error Part 207:40

- GE With Partial Pivoting: Theory10:39

- GE With Partial Pivoting: Example Part 107:15

- GE With Partial Pivoting: Example Part 210:07

- GE With Partial Pivoting: Example Part 306:17

- GE w/ Partial Pivoting: Example of Round Off Error Part 108:58

- GE w/ Partial Pivoting: Example of Round Off Error Part 208:17

- GE w/ Partial Pivoting: Example of Round Off Error Part 305:47

- Determinant of a Matrix Using FE: Background05:17

- Determinant of a Matrix Using FE: Example10:07

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems9 pages

- Textbook Chapter 710 pages

- LU Decomposition Basis09:02

- Finding the Inverse of a Matrix: Theory06:02

- Finding the Inverse of a Matrix: Example10:20

- LU Decoposition: Example Part 106:55

- LU Decoposition: Example Part 104:36

- Solving a Set of Equations: Example10:28

- Advantages of LU Decomposition Part 104:57

- Advantages of LU Decomposition Part 208:05

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems11 pages

- Textbook Chapter 810 pages

- Gauss-Seidel Method: Theory Part 108:00

- Gauss-Seidel Method: Theory Part 205:37

- Gauss-Seidel Method: Example Part 109:16

- Gauss-Seidel Method: Example Part 207:39

- Gauss-Seidel Method: Pitfall Part 107:50

- Gauss-Seidel Method: Pitfall Part 208:13

- End of Chapter Practice Problems1 page

- Solutions to Practice Problems11 pages

- Textbook Chapter 911 pages

- Properties of Norms03:36

- Relation of Norm to the Conditioning of SLEs: Part 108:54

- Relation of Norm to the Conditioning of SLEs: Part 205:57

- Ill-conditioned and Well-conditioned SLEs10:11

- Significant Digits in Solution Vector: Theory03:59

- Significant Digits In Solution Vector: Example 103:56

- Significant Digits In Solution Vector: Example 204:25

- Relate Changes in Coef Matrix to Changes in Soln: Proof08:45

- Relating Changes in Coeff Matrix to Changes in Soln Vec04:17

- Relating Changes in RHS Vec to Changes in Solution Vec03:11

- Row Sum Norm of a Matrix: Example03:05

- Row Sum Norm of a Matrix: Theory Test 202:33

- Row Sum Norm of a Matrix: Theory02:33

- End of Chapter Practice Problems3 pages

- Solutions to Practice Problems9 pages

- Textbook Chapter 1012 pages

- Origin of the Word Eigenvalue01:01

- Theorems of Eigenvalues and Eigenvectors: Part 102:18

- Theorems of Eigenvalues and Eigenvectors: Part 202:05

- Theorems of Eigenvalues and Eigenvectors: Part 302:43

- Theorems of Eigenvalues and Eigenvectors: Part 400:52

- Theorems of Eigenvalues and Eigenvectors: Part 501:36

- Theorems of Eigenvalues and Eigenvectors: Part 603:14

- Definition of Eigenvalues and Eigenvectors03:10

- Eigenvalues of a Square Matrix: Theory04:32

- Eigenvalues of a Square Matrix: Example03:45

- Eigenvectors of a Square Matrix: Example06:32

- Eigenvectors of a Square Matrix: Example 213:09

- Find Eigenvalues and Eigenvectors Numerically: Theory04:56

- Find Eigenvalues and Eigenvectors Numerically: Example08:08

- Application of Eigenvalues and Eigenvectors16:22

- End of Chapter Practice Problems2 pages

- Solutions to Practice Problems7 pages

**Autar Kaw** is a* professo*r 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.