Introduction 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!
0.0 (0 ratings)
59 students enrolled
Created by Autar Kaw
Last updated 6/2015
English
English [Auto-generated]
Price: Free
This course includes
• 11 hours on-demand video
• Access on mobile and TV
What you'll learn
• know vectors and their linear combinations and dot products

• ### 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
Expand all 177 lectures 14:56:03
+ Introduction to Matrices and Vectors
13 lectures 27:46

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.

Textbook Chapter 1
8 pages

This video lecture answers the question: "What is a matrix?"

Definition of a Matrix
02:21

This video lecture answers the question: "What is a square matrix?"

Definition of a Square Matrix
01:43

This video lecture answers the question: "What is a submatrix?"

Definition of a Submatrix
02:43

This video lecture answers the question: "What is a diagonal matrix?"

Diagonal Matrix
02:45

This video lecture defines a diagonally dominant matrix.

Diagonally Dominant Matrix
07:21

This video lecture defines an identity matrix.

Identity Matrix
02:01

This video lecture answers the question: "What is a lower triangular matrix?"

Lower Triangular Matrix
03:13

This video lecture defines what makes two matrices equal.

Equal Matrices
02:53

This video lecture answers the question: "What is a column vector?"

Column Vector
01:23

This video lecture answers the question: "What is a row vector?"

Row Vector
01:23

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
8 pages
+ Vectors
29 lectures 01:10:16

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.

Textbook Chapter 2
16 pages

This video answers the question: "What is a vector?"

Definition of a Vector: Theory
02:26

This video gives an example of a vector.

Definition of a Vector: Example
01:14

This video lecture answers the question: "How do you add two vectors?"

01:33

This video lecture gives an example in vector addition addressed in the previous lecture.

01:47

This video lecture answers the question: "How do you multiply a vector by a scalar?"

Multiply a Vector By a Scalar: Theory
01:17

This video lecture gives an example of the product of a vector and a scalar which was discussed in the previous lecture.

Multiply a Vector By a Scalar: Example
01:15

This video lecture answers the question: "What is the definition of the dot product of two vectors?"

Dot Product: Theory
01:59

This video lecture gives an example of the dot product of two vectors as discussed in the previous lecture.

Dot Product: Example
01:40

This video lecture answers the question: "What is the rank of a set of vectors?"

Rank of a Set of Vectors: Theory
02:08

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 1
03:52

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 2
02:26

This video lecture answers the question: "What is meant by a linear combination of vectors?"

Linear Combination of Vectors: Theory
01:43

This video lecture gives an example of a linear combination of vectors.

Linear Combination of Vectors: Example
02:33

This video lecture answers the question: "How can vectors be used to write simultaneous linear equations?"

Simultaneous Linear Equations in Vector Form: Theory
05:07

This video lecture gives an example of writing simultaneous linear equations in vector form.

Simultaneous Linear Equations in Vector Form: Example
03:12

This video lecture answers the question: "What is a null or zero vector?"

Null or Zero Vectors: Theory
01:11

This video lecture answers the question: "What is a unit vector?"

Unit Vectors: Theory
01:29

This video lecture gives an example of a unit vector.

Unit Vectors: Example
01:48

This video lecture answers the question: "When are two vectors equal?"

Equivalent Vectors: Theory
02:01

This video lecture gives an example of two equal vectors.

Equivalent Vectors: Example
02:25

This lecture will demonstrate that if a set of vectors contains a null vector, then the vectors are linearly dependent.

Linearly Dependent Vectors: Proof 1
02:28

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 Dependent Vectors: Proof 2
04:06

This video lecture answers the question: "What is meant by vectors being linearly independent?"

Linearly Independent Vectors: Theory
02:27

This video lecture gives an example of linearly independent vectors as discussed in the previous lecture.

Linearly Independent Vectors: Example 1
03:16

This video lecture gives an example of linearly independent vectors as discussed in the previous lecture.

Linearly Independent Vectors: Example 2
09:12

This lecture demonstrates that if a set of vectors is linearly independent, then a subset of it is also linearly independent.

Subsets of Linearly Independent Vectors: Proof
05:41

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
7 pages
+ Matrices & Binary
18 lectures 40:39

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.

Textbook Chapter 3
9 pages

This video lecture discusses matrix addition.

01:53

This video lecture gives an example of matrix addition.

02:10

This video lecture discusses matrix subtraction.

Matrix Subtraction: Theory
01:39

This video lecture gives an example of matrix subtraction.

Matrix Subtraction: Example
02:04

This video lecture discusses the linear combination of matrices.

Linear Combination of Matrices: Theory
02:03

This video lecture gives an example of a linear combination of matrices.

Linear Combination of Matrices: Example
03:56

This video lecture discusses matrix multiplication.

Matrix Multiplication: Theory
04:32

This video lecture gives an example of matrix multiplication as discussed in the previous lecture.

Matrix Multiplication: Example
06:19

This video lecture answers the question: "Is matrix multiplication commutative?"

Is Matrix Multiplication Commutative?
04:00

This video lecture discusses the theory behind a product of a scalar and a matrix.

Product of a Scalar and a Matrix: Theory
01:36

This video lecture gives an example of the product of a scalar and a matrix as discussed in the previous lecture.

Product of a Scalar and a Matrix: Example
01:44

This video lecture begins a discussion of the rules associated with binary matrix operations.

Rules of Binary Matrix Operations: Part 1
01:46

This video lecture continues the discussion of the rules associated with binary matrix operations.

Rules of Binary Matrix Operations: Part 2
01:38

This video lecture continues the discussion of the rules associated with binary matrix operations.

Rules of Binary Matrix Operations: Part 3
02:49

This video lecture continues the discussion of the rules associated with binary matrix operations.

Rules of Binary Matrix Operations: Part 4
02:30

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
6 pages
+ Unary
15 lectures 43:53

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.

Textbook Chapter 4
10 pages

This video lecture shows how the determinant of a matrix can be found using cofactors.

Determinant of a Matrix Using Cofactors: Theory
03:25

This video lecture gives an example of finding the determinant of a matrix using cofactors.

Determinant of a Matrix Using Cofactors: Example
05:30

This video lecture discusses finding the determinant of a matrix using minors.

Determinant of a Matrix Using Minors: Theory
04:40

This video lecture gives an example using minors to find the determinant of a matrix.

Determinant of a Matrix Using Minors: Example
06:25

This video lecture discusses what a skew-symmetric matrix is.

Skew-symmetric Matrix
03:11

This video lecture discusses what a symmetric matrix is.

Symmetric Matrix
03:19

This video lecture begins a discussion about the theorems on determinants.

Theorems on Determinants: Part 1
01:35

This video lecture continues a discussion about the theorems on determinants.

Theorems on Determinants: Part 2
03:34

This video lecture continues a discussion about the theorems on determinants.

Theorems on Determinants: Part 3
03:20

This video lecture continues a discussion about the theorems on determinants.

Theorems on Determinants: Part 4
02:41

This video lecture discusses the trace of a matrix.

Trace of a Matrix
02:02

This video lecture discusses the transpose of a matrix.

Transpose of a Matrix
04:11

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
6 pages
+ System of Equations
28 lectures 01:46:15

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.

Textbook Chapter 5
18 pages

This video lecture discusses how to write simultaneous linear equations in matrix form.

Writing Simultaneous Linear Equations in Matrix Form
05:25

This video lecture gives an example of a real life problem of setting up simultaneous linear equations.

Setting Up Simultaneous Linear Equations: Example
05:22

This video lecture answers the question: "Can a system of equations have more than one solution?"

Number of Solutions for a System of Linear Equations
04:56

This video lecture discusses consistent and inconsistent systems of equations.

Consistent and Inconsistent System of Equations: Theory
02:55

This video lecture gives examples of consistent and inconsistent systems of equations as discussed in the previous lecture.

Consistent and Inconsistent System of Equations: Example
05:30

This video lecture discusses how to distinguish between consistent and inconsistent systems of equations.

Consistent and Inconsistent System of Equations
03:05

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 1
04:14

This video lecture gives an example of distinguishing between consistent and inconsistent systems of equations as previously discussed.

Consistent and Inconsistent System of Equations: Example 2
08:41

This video lecture discusses how to determine the uniqueness of a solution.

Determining the Uniqueness of a Solution
03:24

This video lecture gives an example of distinguishing between consistent and inconsistent systems of equations as previously discussed.

Consistent and Inconsistent System of Equations: Example 3
06:06

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 1
02:17

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 2
02:22

This video lecture answers the question: "Can we divide two matrices?"

Matrix Division
05:38

This video lecture discusses finding the inverse of a matrix.

Finding the Inverse of a Matrix: Theory
04:31

This video lecture gives an example of finding the inverse of a matrix.

Finding the Inverse of a Matrix: Example
07:02

This video lecture discusses finding the inverse of a matrix by adjoints.

Finding the Inverse of a Matrix by Adjoints: Theory
02:14

This video lecture gives an example of finding the inverse of a matrix by adjoints.

Finding the Inverse of a Matrix by Adjoints: Example
07:18

This video lecture answers the question: "If the inverse of a matrix exists, is it unique?"

Uniqueness of a Matrix
02:29

This video lecture answers the question: "If we have more equations than unknowns, does it mean we have inconsistent system of equations?"

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

This video lecture gives an example of taking the inverse of a matrix.

Inverse of Matrices: Example
03:39

This video lecture discusses how to find the rank of a matrix.

Rank of a Matrix: Theory
01:20

This video lecture gives an example of the rank of a matrix.

Rank of a Matrix: Example 1
01:30

The video lecture gives an example of the rank of a matrix.

Rank of a Matrix: Example 2
02:49

This video lecture gives some statements about the inverse of matrices.

Facts About the Inverse of a Matrix
02:52

This video lecture demonstrates how the inverse of a matrix can be used to solve a set of equations.

Solving a Set of Equations With the Inverse
02:26

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!

End of Chapter Practice Problems
3 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
10 pages
+ Gaussian Elimination (GE)
20 lectures 02:13:29

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.

Textbook Chapter 6
18 pages

This video lecture begins a discussion of Naive Gaussian Elimination.

Naive GE: Theory Part 1
10:27

This video lecture continues the discussion of Naive Gaussian Elimination form the previous lecture.

Naive GE: Theory Part 2
02:22

This video lecture gives an example of Naive Gaussian Elimination using Forward Elimination.

Naive GE: Example 1 Part 1 Forward Elimination
10:49

This video lecture gives an example of Naive Gaussian Elimination using Backward Substitution.

Naive GE: Example 2 Part 1 Backward Substitution
08:07

This video lecture continues an example of Naive Gaussian Elimination using Backward Substitution which was begun in the previous lecture.

Naive GE: Example 2 Part 2 Backward Substitution
06:40

This video lecture discusses the problems associated with Naive Gaussian Elimination.

Naive GE: Pitfalls
07:20

This video lecture begins an example of the round off error associated with Naive Gaussian Elimination.

Naive GE: Example of Round Off Error Part 1
07:20

This video lecture continues an example of the round off error associated with Naive Gaussian Elimination which was begun in the previous lecture.

Naive GE: Example of Round Off Error Part 2
07:40

This video lecture discusses the theory behind Gaussian Elimination With Partial Pivoting.

GE With Partial Pivoting: Theory
10:39

This video lecture is part 1 of an example of Gaussian Elimination With Partial Pivoting.

GE With Partial Pivoting: Example Part 1
07:15

This video lecture is part 2 of an example of Gaussian Elimination With Partial Pivoting.

GE With Partial Pivoting: Example Part 2
10:07

This video lecture is part 3 of an example of Gaussian Elimination With Partial Pivoting.

GE With Partial Pivoting: Example Part 3
06:17

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 1
08:58

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 2
08:17

This video lecture is part 3 of an example of round off error associated with Gaussian Elimination With Partial Pivoting.

GE w/ Partial Pivoting: Example of Round Off Error Part 3
05:47

This video lecture describes finding the determinant of a matrix using forward elimination.

Determinant of a Matrix Using FE: Background
05:17

This video lecture gives an example of finding the determinant of a matrix using forward elimination.

Determinant of a Matrix Using FE: Example
10:07

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
9 pages
+ LU Decomposition
11 lectures 01:00:25

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.

Textbook Chapter 7
10 pages

This video lecture discusses the basis of the LU Decomposition method.

LU Decomposition Basis
09:02

This video lecture discusses the theory behind finding the inverse of a matrix.

Finding the Inverse of a Matrix: Theory
06:02

This video lecture gives an example of finding the inverse of a matrix.

Finding the Inverse of a Matrix: Example
10:20

This video lecture is part 1 of an example on using LU Decomposition.

LU Decoposition: Example Part 1
06:55

This video lecture is part 2 of an example on using LU Decomposition.

LU Decoposition: Example Part 1
04:36

This video lecture shows how to use LU Decomposition to solve a set of equations.

Solving a Set of Equations: Example
10:28

This video lecture is part 1 of a discussion on the advantages of using LU Decomposition.

Advantages of LU Decomposition Part 1
04:57

This video lecture is part 2 of a discussion on the advantages of using LU Decomposition.

Advantages of LU Decomposition Part 2
08:05

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
11 pages
+ Gauss-Seidel Method
9 lectures 46:35

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.

Textbook Chapter 8
10 pages

This video lecture is part 1 of a discussion on theory behind the Gauss-Seidel Method.

Gauss-Seidel Method: Theory Part 1
08:00

This video lecture is part 2 of a discussion on theory behind the Gauss-Seidel Method.

Gauss-Seidel Method: Theory Part 2
05:37

This video lecture is part 1 of an example using the Gauss-Seidel Method.

Gauss-Seidel Method: Example Part 1
09:16

This video lecture is part 2 of an example using the Gauss-Seidel Method.

Gauss-Seidel Method: Example Part 2
07:39

This video lecture is part 1 of a discussion on the pitfalls associated with the Gauss-Seidel method.

Gauss-Seidel Method: Pitfall Part 1
07:50

This video lecture is part 2 of a discussion on the pitfalls associated with the Gauss-Seidel method.

Gauss-Seidel Method: Pitfall Part 2
08:13

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!

End of Chapter Practice Problems
1 page

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
11 pages
16 lectures 01:05:22

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.

Textbook Chapter 9
11 pages

This video lecture discusses the properties of norms.

Properties of Norms
03:36

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 1
08:54

This video lecture is part 2 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 2
05:57

This video lecture discusses whether a system of equations is i-ll conditioned or well-conditioned.

Ill-conditioned and Well-conditioned SLEs
10:11

This video lecture discusses the theory behind the number of significant digits that is correct in a solution vector.

Significant Digits in Solution Vector: Theory
03:59

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 1
03:56

This video lecture gives another example of the number of significant digits that is correct in a solution vector.

Significant Digits In Solution Vector: Example 2
04:25

This video lecture explains the proof of how changes in coefficient matrix are related to changes in solution vector.

Relate Changes in Coef Matrix to Changes in Soln: Proof
08:45

This video lecture discusses relating changes in coefficient matrix to changes in solution vector.

Relating Changes in Coeff Matrix to Changes in Soln Vec
04:17

This video lecture discusses relating changes in right hand side vector to changes in solution vector.

Relating Changes in RHS Vec to Changes in Solution Vec
03:11

This video lecture gives an example of a row sum norm of a matrix.

Row Sum Norm of a Matrix: Example
03:05

This video lecture continues the discussion on the theory behind the row sum norm of a matrix.

Row Sum Norm of a Matrix: Theory Test 2
02:33

This video lecture explains the theory behind the row sum norm of a matrix.

Row Sum Norm of a Matrix: Theory
02:33

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!

End of Chapter Practice Problems
3 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
9 pages
+ Eigenvalues and Eigenvectors
18 lectures 01:14:23

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.

Textbook Chapter 10
12 pages

This video explains the origin of the word "eigenvalue".

Origin of the Word Eigenvalue
01:01

This video lecture is part 1 of a discussion on the theorems of eigenvalues and eigenvectors.

Theorems of Eigenvalues and Eigenvectors: Part 1
02:18

This video lecture is part 2 of a discussion on the theorems of eigenvalues and eigenvectors.

Theorems of Eigenvalues and Eigenvectors: Part 2
02:05

This video lecture is part 3 of a discussion on the theorems of eigenvalues and eigenvectors.

Theorems of Eigenvalues and Eigenvectors: Part 3
02:43

This video lecture is part 4 of a discussion on the theorems of eigenvalues and eigenvectors.

Theorems of Eigenvalues and Eigenvectors: Part 4
00:52

This video lecture is part 5 of a discussion on the theorems of eigenvalues and eigenvectors.

Theorems of Eigenvalues and Eigenvectors: Part 5
01:36

This video lecture is part 6 of a discussion on the theorems of eigenvalues and eigenvectors.

Theorems of Eigenvalues and Eigenvectors: Part 6
03:14

This video lecture defines what eigenvalues and eigenvectors are.

Definition of Eigenvalues and Eigenvectors
03:10

This video lecture discusses the theory of how to find the eigenvalues of a square matrix.

Eigenvalues of a Square Matrix: Theory
04:32

This video lecture gives an example of finding the eigenvalues of a square matrix.

Eigenvalues of a Square Matrix: Example
03:45

This video lecture gives an example of finding the eigenvectors of a square matrix.

Eigenvectors of a Square Matrix: Example
06:32

This video lecture gives another example of finding the eigenvectors of a square matrix.

Eigenvectors of a Square Matrix: Example 2
13:09

This video lecture discusses the theory behind finding the eigenvalues and eigenvectors numerically.

Find Eigenvalues and Eigenvectors Numerically: Theory
04:56

This video lecture gives an example of finding the eigenvalues and eigenvectors numerically.

Find Eigenvalues and Eigenvectors Numerically: Example
08:08

This video lecture gives a physical example of the application of eigenvalues and eigevectors.

Application of Eigenvalues and Eigenvectors
16:22

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!

End of Chapter Practice Problems
2 pages

The solutions to the practice problems are provided here for you to check your approach and answers.

Solutions to Practice Problems
7 pages