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Learn MoreIntroduction to Matrix Algebra
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Introduction to Matrix Algebra
A just-in-time tool for various STEM courses and a much needed refresher!
Created byAutar Kaw
Last updated 6/2015
English
What you'll learn
- 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
Explore related topics
Course content
10 sections • 177 lectures • 11h 9m total length
- Textbook Chapter 18:00
This is the textbook chapter for the Section 1: Introduction to Matrices and Vectors. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Definition of a Matrix2:21
This video lecture answers the question: "What is a matrix?"
- Definition of a Square Matrix1:43
This video lecture answers the question: "What is a square matrix?"
- Definition of a Submatrix2:43
This video lecture answers the question: "What is a submatrix?"
- Diagonal Matrix2:45
This video lecture answers the question: "What is a diagonal matrix?"
- Diagonally Dominant Matrix7:21
This video lecture defines a diagonally dominant matrix.
- Identity Matrix2:01
This video lecture defines an identity matrix.
- Lower Triangular Matrix3:13
This video lecture answers the question: "What is a lower triangular matrix?"
- Equal Matrices2:53
This video lecture defines what makes two matrices equal.
- Column Vector1:23
This video lecture answers the question: "What is a column vector?"
- Row Vector1:23
This video lecture answers the question: "What is a row vector?"
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems8:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 216:00
This is the textbook chapter for the Section 2: Vectors. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Definition of a Vector: Theory2:26
This video answers the question: "What is a vector?"
- Definition of a Vector: Example1:14
This video gives an example of a vector.
- Vector Addition: Theory1:33
This video lecture answers the question: "How do you add two vectors?"
- Vector Addition: Example1:47
This video lecture gives an example in vector addition addressed in the previous lecture.
- Multiply a Vector By a Scalar: Theory1:17
This video lecture answers the question: "How do you multiply a vector by a scalar?"
- Multiply a Vector By a Scalar: Example1:15
This video lecture gives an example of the product of a vector and a scalar which was discussed in the previous lecture.
- Dot Product: Theory1:59
This video lecture answers the question: "What is the definition of the dot product of two vectors?"
- Dot Product: Example1:40
This video lecture gives an example of the dot product of two vectors as discussed in the previous lecture.
- Rank of a Set of Vectors: Theory2:08
This video lecture answers the question: "What is the rank of a set of vectors?"
- Rank of a Set of Vectors: Example 13:52
This video lecture gives an example of the rank of a set of vectors as discussed in the previous lecture.
- Rank of a Set of Vectors: Example 22:26
This video lecture gives an example of the rank of a set of vectors as discussed in the previous lecture.
- Linear Combination of Vectors: Theory1:43
This video lecture answers the question: "What is meant by a linear combination of vectors?"
- Linear Combination of Vectors: Example2:33
This video lecture gives an example of a linear combination of vectors.
- Simultaneous Linear Equations in Vector Form: Theory5:07
This video lecture answers the question: "How can vectors be used to write simultaneous linear equations?"
- Simultaneous Linear Equations in Vector Form: Example3:12
This video lecture gives an example of writing simultaneous linear equations in vector form.
- Null or Zero Vectors: Theory1:11
This video lecture answers the question: "What is a null or zero vector?"
- Unit Vectors: Theory1:29
This video lecture answers the question: "What is a unit vector?"
- Unit Vectors: Example1:48
This video lecture gives an example of a unit vector.
- Equivalent Vectors: Theory2:01
This video lecture answers the question: "When are two vectors equal?"
- Equivalent Vectors: Example2:25
This video lecture gives an example of two equal vectors.
- Linearly Dependent Vectors: Proof 12:28
This lecture will demonstrate that if a set of vectors contains a null vector, then the vectors are linearly dependent.
- Linearly Dependent Vectors: Proof 24:06
This lecture will demonstrate that if a set of vectors is linearly dependent, then at least one of the vectors can be written as a linear combination of the others.
- Linearly Independent Vectors: Theory2:27
This video lecture answers the question: "What is meant by vectors being linearly independent?"
- Linearly Independent Vectors: Example 13:16
This video lecture gives an example of linearly independent vectors as discussed in the previous lecture.
- Linearly Independent Vectors: Example 29:12
This video lecture gives an example of linearly independent vectors as discussed in the previous lecture.
- Subsets of Linearly Independent Vectors: Proof5:41
This lecture demonstrates that if a set of vectors is linearly independent, then a subset of it is also linearly independent.
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems7:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 39:00
This is the textbook chapter for the Section 3: Binary. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Matrix Addition: Theory1:53
This video lecture discusses matrix addition.
- Matrix Addition: Example2:10
This video lecture gives an example of matrix addition.
- Matrix Subtraction: Theory1:39
This video lecture discusses matrix subtraction.
- Matrix Subtraction: Example2:04
This video lecture gives an example of matrix subtraction.
- Linear Combination of Matrices: Theory2:03
This video lecture discusses the linear combination of matrices.
- Linear Combination of Matrices: Example3:56
This video lecture gives an example of a linear combination of matrices.
- Matrix Multiplication: Theory4:32
This video lecture discusses matrix multiplication.
- Matrix Multiplication: Example6:19
This video lecture gives an example of matrix multiplication as discussed in the previous lecture.
- Is Matrix Multiplication Commutative?4:00
This video lecture answers the question: "Is matrix multiplication commutative?"
- Product of a Scalar and a Matrix: Theory1:36
This video lecture discusses the theory behind a product of a scalar and a matrix.
- Product of a Scalar and a Matrix: Example1:44
This video lecture gives an example of the product of a scalar and a matrix as discussed in the previous lecture.
- Rules of Binary Matrix Operations: Part 11:46
This video lecture begins a discussion of the rules associated with binary matrix operations.
- Rules of Binary Matrix Operations: Part 21:38
This video lecture continues the discussion of the rules associated with binary matrix operations.
- Rules of Binary Matrix Operations: Part 32:49
This video lecture continues the discussion of the rules associated with binary matrix operations.
- Rules of Binary Matrix Operations: Part 42:30
This video lecture continues the discussion of the rules associated with binary matrix operations.
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems6:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 410:00
This is the textbook chapter for the Section 4: Unary. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Determinant of a Matrix Using Cofactors: Theory3:25
This video lecture shows how the determinant of a matrix can be found using cofactors.
- Determinant of a Matrix Using Cofactors: Example5:30
This video lecture gives an example of finding the determinant of a matrix using cofactors.
- Determinant of a Matrix Using Minors: Theory4:40
This video lecture discusses finding the determinant of a matrix using minors.
- Determinant of a Matrix Using Minors: Example6:25
This video lecture gives an example using minors to find the determinant of a matrix.
- Skew-symmetric Matrix3:11
This video lecture discusses what a skew-symmetric matrix is.
- Symmetric Matrix3:19
This video lecture discusses what a symmetric matrix is.
- Theorems on Determinants: Part 11:35
This video lecture begins a discussion about the theorems on determinants.
- Theorems on Determinants: Part 23:34
This video lecture continues a discussion about the theorems on determinants.
- Theorems on Determinants: Part 33:20
This video lecture continues a discussion about the theorems on determinants.
- Theorems on Determinants: Part 42:41
This video lecture continues a discussion about the theorems on determinants.
- Trace of a Matrix2:02
This video lecture discusses the trace of a matrix.
- Transpose of a Matrix4:11
This video lecture discusses the transpose of a matrix.
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems6:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 518:00
This is the textbook chapter for the Section 5: System of Equations. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Writing Simultaneous Linear Equations in Matrix Form5:25
This video lecture discusses how to write simultaneous linear equations in matrix form.
- Setting Up Simultaneous Linear Equations: Example5:22
This video lecture gives an example of a real life problem of setting up simultaneous linear equations.
- Number of Solutions for a System of Linear Equations4:56
This video lecture answers the question: "Can a system of equations have more than one solution?"
- Consistent and Inconsistent System of Equations: Theory2:55
This video lecture discusses consistent and inconsistent systems of equations.
- Consistent and Inconsistent System of Equations: Example5:30
This video lecture gives examples of consistent and inconsistent systems of equations as discussed in the previous lecture.
- Consistent and Inconsistent System of Equations3:05
This video lecture discusses how to distinguish between consistent and inconsistent systems of equations.
- Consistent and Inconsistent System of Equations: Example 14:14
This video lecture gives an example of distinguishing between consistent and inconsistent systems of equations as discussed in the previous lecture.
- Consistent and Inconsistent System of Equations: Example 28:41
This video lecture gives an example of distinguishing between consistent and inconsistent systems of equations as previously discussed.
- Determining the Uniqueness of a Solution3:24
This video lecture discusses how to determine the uniqueness of a solution.
- Consistent and Inconsistent System of Equations: Example 36:06
This video lecture gives an example of distinguishing between consistent and inconsistent systems of equations as previously discussed.
- Does a Set of Equations Have a Unique Solution: Example 12:17
This video lecture gives an example of determining whether a set of equations has a unique solution or not.
- Does a Set of Equations Have a Unique Solution: Example 22:22
This video lecture gives an example of determining whether a set of equations has a unique solution or not.
- Matrix Division5:38
This video lecture answers the question: "Can we divide two matrices?"
- Finding the Inverse of a Matrix: Theory4:31
This video lecture discusses finding the inverse of a matrix.
- Finding the Inverse of a Matrix: Example7:02
This video lecture gives an example of finding the inverse of a matrix.
- Finding the Inverse of a Matrix by Adjoints: Theory2:14
This video lecture discusses finding the inverse of a matrix by adjoints.
- Finding the Inverse of a Matrix by Adjoints: Example7:18
This video lecture gives an example of finding the inverse of a matrix by adjoints.
- Uniqueness of a Matrix2:29
This video lecture answers the question: "If the inverse of a matrix exists, is it unique?"
- Does more than one unknown mean inconsistent equations?8:10
This video lecture answers the question: "If we have more equations than unknowns, does it mean we have inconsistent system of equations?"
- Inverse of Matrices: Example3:39
This video lecture gives an example of taking the inverse of a matrix.
- Rank of a Matrix: Theory1:20
This video lecture discusses how to find the rank of a matrix.
- Rank of a Matrix: Example 11:30
This video lecture gives an example of the rank of a matrix.
- Rank of a Matrix: Example 22:49
The video lecture gives an example of the rank of a matrix.
- Facts About the Inverse of a Matrix2:52
This video lecture gives some statements about the inverse of matrices.
- Solving a Set of Equations With the Inverse2:26
This video lecture demonstrates how the inverse of a matrix can be used to solve a set of equations.
- End of Chapter Practice Problems3:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems10:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 618:00
This is the textbook chapter for the Section 6: Gaussian Elimination. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Naive GE: Theory Part 110:27
This video lecture begins a discussion of Naive Gaussian Elimination.
- Naive GE: Theory Part 22:22
This video lecture continues the discussion of Naive Gaussian Elimination form the previous lecture.
- Naive GE: Example 1 Part 1 Forward Elimination10:49
This video lecture gives an example of Naive Gaussian Elimination using Forward Elimination.
- Naive GE: Example 2 Part 1 Backward Substitution8:07
This video lecture gives an example of Naive Gaussian Elimination using Backward Substitution.
- Naive GE: Example 2 Part 2 Backward Substitution6:40
This video lecture continues an example of Naive Gaussian Elimination using Backward Substitution which was begun in the previous lecture.
- Naive GE: Pitfalls7:20
This video lecture discusses the problems associated with Naive Gaussian Elimination.
- Naive GE: Example of Round Off Error Part 17:20
This video lecture begins an example of the round off error associated with Naive Gaussian Elimination.
- Naive GE: Example of Round Off Error Part 27:40
This video lecture continues an example of the round off error associated with Naive Gaussian Elimination which was begun in the previous lecture.
- GE With Partial Pivoting: Theory10:39
This video lecture discusses the theory behind Gaussian Elimination With Partial Pivoting.
- GE With Partial Pivoting: Example Part 17:15
This video lecture is part 1 of an example of Gaussian Elimination With Partial Pivoting.
- GE With Partial Pivoting: Example Part 210:07
This video lecture is part 2 of an example of Gaussian Elimination With Partial Pivoting.
- GE With Partial Pivoting: Example Part 36:17
This video lecture is part 3 of an example of Gaussian Elimination With Partial Pivoting.
- GE w/ Partial Pivoting: Example of Round Off Error Part 18:58
This video lecture is part 1 of an example of round off error associated with Gaussian Elimination With Partial Pivoting.
- GE w/ Partial Pivoting: Example of Round Off Error Part 28:17
This video lecture is part 2 of an example of round off error associated with Gaussian Elimination With Partial Pivoting.
- GE w/ Partial Pivoting: Example of Round Off Error Part 35:47
This video lecture is part 3 of an example of round off error associated with Gaussian Elimination With Partial Pivoting.
- Determinant of a Matrix Using FE: Background5:17
This video lecture describes finding the determinant of a matrix using forward elimination.
- Determinant of a Matrix Using FE: Example10:07
This video lecture gives an example of finding the determinant of a matrix using forward elimination.
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems9:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 710:00
This is the textbook chapter for the Section 7: LU Decomposition. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- LU Decomposition Basis9:02
This video lecture discusses the basis of the LU Decomposition method.
- Finding the Inverse of a Matrix: Theory6:02
This video lecture discusses the theory behind finding the inverse of a matrix.
- Finding the Inverse of a Matrix: Example10:20
This video lecture gives an example of finding the inverse of a matrix.
- LU Decoposition: Example Part 16:55
This video lecture is part 1 of an example on using LU Decomposition.
- LU Decoposition: Example Part 14:36
This video lecture is part 2 of an example on using LU Decomposition.
- Solving a Set of Equations: Example10:28
This video lecture shows how to use LU Decomposition to solve a set of equations.
- Advantages of LU Decomposition Part 14:57
This video lecture is part 1 of a discussion on the advantages of using LU Decomposition.
- Advantages of LU Decomposition Part 28:05
This video lecture is part 2 of a discussion on the advantages of using LU Decomposition.
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems11:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 810:00
This is the textbook chapter for the Section 8: Gauss-Seidel Method for System of Linear Equations. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Gauss-Seidel Method: Theory Part 18:00
This video lecture is part 1 of a discussion on theory behind the Gauss-Seidel Method.
- Gauss-Seidel Method: Theory Part 25:37
This video lecture is part 2 of a discussion on theory behind the Gauss-Seidel Method.
- Gauss-Seidel Method: Example Part 19:16
This video lecture is part 1 of an example using the Gauss-Seidel Method.
- Gauss-Seidel Method: Example Part 27:39
This video lecture is part 2 of an example using the Gauss-Seidel Method.
- Gauss-Seidel Method: Pitfall Part 17:50
This video lecture is part 1 of a discussion on the pitfalls associated with the Gauss-Seidel method.
- Gauss-Seidel Method: Pitfall Part 28:13
This video lecture is part 2 of a discussion on the pitfalls associated with the Gauss-Seidel method.
- End of Chapter Practice Problems1:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems11:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 911:00
This is the textbook chapter for the Section 9: Adequacy of Solutions. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Properties of Norms3:36
This video lecture discusses the properties of norms.
- Relation of Norm to the Conditioning of SLEs: Part 18:54
This video lecture is part 1 of answering the question: "How is the norm related to the conditioning of a system of equations?"
- Relation of Norm to the Conditioning of SLEs: Part 25:57
This video lecture is part 2 of answering the question: "How is the norm related to the conditioning of a system of equations?"
- Ill-conditioned and Well-conditioned SLEs10:11
This video lecture discusses whether a system of equations is i-ll conditioned or well-conditioned.
- Significant Digits in Solution Vector: Theory3:59
This video lecture discusses the theory behind the number of significant digits that is correct in a solution vector.
- Significant Digits In Solution Vector: Example 13:56
This video lecture gives an example of the number of significant digits that is correct in a solution vector.
- Significant Digits In Solution Vector: Example 24:25
This video lecture gives another example of the number of significant digits that is correct in a solution vector.
- Relate Changes in Coef Matrix to Changes in Soln: Proof8:45
This video lecture explains the proof of how changes in coefficient matrix are related to changes in solution vector.
- Relating Changes in Coeff Matrix to Changes in Soln Vec4:17
This video lecture discusses relating changes in coefficient matrix to changes in solution vector.
- Relating Changes in RHS Vec to Changes in Solution Vec3:11
This video lecture discusses relating changes in right hand side vector to changes in solution vector.
- Row Sum Norm of a Matrix: Example3:05
This video lecture gives an example of a row sum norm of a matrix.
- Row Sum Norm of a Matrix: Theory Test 22:33
This video lecture continues the discussion on the theory behind the row sum norm of a matrix.
- Row Sum Norm of a Matrix: Theory2:33
This video lecture explains the theory behind the row sum norm of a matrix.
- End of Chapter Practice Problems3:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems9:00
The solutions to the practice problems are provided here for you to check your approach and answers.
- Textbook Chapter 1012:00
This is the textbook chapter for the Section 10: Eigenvalues and Eigenvectors. Here you will find the examples and theory that can be seen in the following video lectures. For practice problems, see the End of Chapter Problems for this section.
- Origin of the Word Eigenvalue1:01
This video explains the origin of the word "eigenvalue".
- Theorems of Eigenvalues and Eigenvectors: Part 12:18
This video lecture is part 1 of a discussion on the theorems of eigenvalues and eigenvectors.
- Theorems of Eigenvalues and Eigenvectors: Part 22:05
This video lecture is part 2 of a discussion on the theorems of eigenvalues and eigenvectors.
- Theorems of Eigenvalues and Eigenvectors: Part 32:43
This video lecture is part 3 of a discussion on the theorems of eigenvalues and eigenvectors.
- Theorems of Eigenvalues and Eigenvectors: Part 40:52
This video lecture is part 4 of a discussion on the theorems of eigenvalues and eigenvectors.
- Theorems of Eigenvalues and Eigenvectors: Part 51:36
This video lecture is part 5 of a discussion on the theorems of eigenvalues and eigenvectors.
- Theorems of Eigenvalues and Eigenvectors: Part 63:14
This video lecture is part 6 of a discussion on the theorems of eigenvalues and eigenvectors.
- Definition of Eigenvalues and Eigenvectors3:10
This video lecture defines what eigenvalues and eigenvectors are.
- Eigenvalues of a Square Matrix: Theory4:32
This video lecture discusses the theory of how to find the eigenvalues of a square matrix.
- Eigenvalues of a Square Matrix: Example3:45
This video lecture gives an example of finding the eigenvalues of a square matrix.
- Eigenvectors of a Square Matrix: Example6:32
This video lecture gives an example of finding the eigenvectors of a square matrix.
- Eigenvectors of a Square Matrix: Example 213:09
This video lecture gives another example of finding the eigenvectors of a square matrix.
- Find Eigenvalues and Eigenvectors Numerically: Theory4:56
This video lecture discusses the theory behind finding the eigenvalues and eigenvectors numerically.
- Find Eigenvalues and Eigenvectors Numerically: Example8:08
This video lecture gives an example of finding the eigenvalues and eigenvectors numerically.
- Application of Eigenvalues and Eigenvectors16:22
This video lecture gives a physical example of the application of eigenvalues and eigevectors.
- End of Chapter Practice Problems2:00
These practice problems can provide you with a quiz to test your knowledge, or they may serve as more examples. The solutions have been provided so you may choose whichever is best for you!
- Solutions to Practice Problems7:00
The solutions to the practice problems are provided here for you to check your approach and answers.
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!