Linear Algebra II
Purpose of Course showclose
Linear algebra is the study of vector spaces and linear mappings between them. In this course, we will begin by reviewing topics you learned in Linear Algebra I, starting with linear equations, followed by a review of vectors and matrices in the context of linear equations. The review will refresh your knowledge of the fundamentals of vectors and of matrix theory, how to perform operations on matrices, and how to solve systems of equations. After the review, you should be able to understand complex numbers from algebraic and geometric viewpoints to the fundamental theorem of algebra. Next, we will focus on eigenvalues and eigenvectors. Today, these have applications in such diverse fields as computer science (Google’s PageRank algorithm), physics (quantum mechanics, vibration analysis, etc.), economics (equilibrium states of Markov models), and more. We will end with the spectral theorem, which provides a decomposition of the vector space on which operators act, and singularvalue decomposition, which is a generalization of the spectral theorem to arbitrary matrices. Then, we will study vector spaces: real, complex, and abstract (i.e., vector space of dimension N over an arbitrary field K) linear transformations. Vector spaces are structures formed by a collection of vectors and are characterized by their dimensions. We will then introduce a new structure on vector spaces: an inner product. Inner products allow us to introduce geometric aspects, such as length of a vector, and to define the notion of orthogonality between vectors. In this context, we will study the geometric aspects of linear algebra by using Euclidean spaces as a guide. If you encounter a theorem that seems difficult or does not seem intuitive, try to study that theorem in the simplest case possible and then move on to more abstract cases. For example, if you are uncomfortable with abstract vector spaces (V) over an arbitrary field (K), then you can fall back on intuition from such spaces as R and C (real and complex). Alternatively, you can reduce the dimension of the vector spaces involved as many notions can be understood in the twodimensional case.
Course Information showclose
Welcome to MA212. Below, please find general information on the course and its requirements.
Primary Resources: This course is comprised of the following primary materials:
 UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics
 Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will also need to complete the following:
 Subsubunit 1.1.3 Activity
 Subsubunit 1.1.4 Activity
 Subunit 1.2 Activity
 Subsubunit 1.3.1 Activity
 Subsubunit 1.3.2 Activity
 Subunit 1.4 Activity
 Subsubunit 2.1.2 Activities
 Subunit 2.3 Activity
 Subunit 2.4 Activity
 Subsubunit 3.1.2.1 Activity
 Subsubunit 3.1.2.3 Activity
 Subsubunit 3.1.3 Activity
 Subsubunit 3.2.3 Activities
 Subunit 3.3 Activity
 Subunit 3.4 Activity
 Subsubunit 4.1.3 Activities
 Subsubunit 4.2.4 Activities
 The Final Exam
Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through the resources in each unit and the activities listed above.
In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: This course should take you a total of 117.5 hours to complete. This is only an approximation, and the course may take longer to complete. Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit. These should help you plan your time accordingly. It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take you about 32.5 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 11.5 hours) over three days, for example by completing subsubunits 1.1.1, 1.1.2, and half of 1.1.3 (a total of 4.5 hours) on Monday; the second half of subsubunit 1.1.3 and subsubunit 1.1.4 (a total of 5 hours) on Tuesday; and subsubunit 1.1.5 and 1.1.6 (about 2 hours) on Wednesday; etc.
Tips/Suggestions: As noted in the “Course Requirements,” Linear Algebra I is a prerequisite for this course. If you are struggling with the material as you progress through this course, consider taking a break to revisit MA211 Linear Algebra. It will likely be helpful to have a graphing calculator on hand for this course. If you do not own or have access to one, consider using this free graphing calculator. As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, and definitions that stand out to you. These notes will serve as a useful review as you study for the Final Exam.
A version of this course is also available in iTunes U.
Preview the course in your browser or view all our iTunes U courses. 
Learning Outcomes showclose
 Define and apply the abstract notions of vector space and inner product space.
 Identify examples of vector spaces.
 Diagonalize a matrix.
 Formulate what a system of linear equations is in terms of matrices.
 Determine whether or not a space has the Archimedean property.
 Use the polar form and geometric interpretation of the complex numbers to solve problems.
 Explain what the fundamental theorem of algebra states.
 State the Fredholm alternative.
 Compute eigenvalues and eigenvectors.
 State Schur's Theorem.
 Define normal matrices.
 Explain the composition and the inversion of permutations.
 Define and compute the determinant.
 Explain when eigenvalues exist for a given operator.
 Compute the normal form of a nilpotent operator.
 Explain the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix.
 Define and identify quadratic forms.
 State the second derivative test.
 Define and compute eigenvectors and eigenvalues.
 Define a vector space and state its properties.
 Compute the linear span of a set of vectors.
 Determine the linear independence or dependence of a set of vectors.
 Determine a basis of a vector space.
 Explain the ideas of linear independence, spanning set, basis, and dimension.
 Define and identify linear transformations.
 Define and compute the characteristic polynomial of a matrix.
 Define and compute a Markov matrix.
 Identify and compute stochastic matrices.
 Define and identify normed vector spaces.
 Apply the Cauchy Schwarz inequality.
 State the Riesz representation theorem.
 State what it means for a nxn matrix to be diagonalizable.
 Define and identify Hermitian operators.
 Define and identify Hilbert spaces.
 Prove the CayleyHamilton theorem.
 Define and determine the adjoint of an operator.
 Define and identify normal operators.
 State the spectral theorem.
 Explain how to find the singularvalue decomposition of an operator.
 Define the notion of length for abstract vectors in abstract vector spaces.
 Define and identify orthogonal vectors.
 Define and identify orthogonal and orthonormal subsets of R^n.
 Perform rangenullspace decompositions.
 Perform orthogonal decomposition of space.
 Perform singularvalue decomposition.
 Use the GramSchmidt process.
 Read and interpret advanced mathematical definitions, theorems, and proofs.
 Produce wellwritten proofs.
Course Requirements showclose
√ Have access to a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plugins or software (e.g., Adobe Reader or Flash).
√ Have the ability to download and save files and documents to a computer.
√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).
√ Be competent in the English language.
√ Have read the Saylor Student Handbook.
√ Have taken the following course from the “Core Program” of the mathematics discipline as a prerequisite: MA211: Linear Algebra.
Preliminary Information

Preliminary Information
Open Textbook Challenge Winner: Linear Algebra, Theory and Applications
Linear Algebra, Theory and Applications was written and submitted by Dr. Kenneth Kuttler of Brigham Young University. Dr. Kuttler wrote this textbook for use by his students at BYU. According to the preface of the text, “This is a book on linear algebra and matrix theory. While it is selfcontained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however.” A solutions manual to the textbook is included.
Linear Algebra, Theory and Applications (PDF)
Linear Algebra, Theory and Applications Solutions Manual (PDF)
Unit Outline show close
Expand All Resources Collapse All Resources

Unit 1: The Basics of Linear Algebra
This unit serves as a review of some of the material covered in Linear Algebra I, including linear equations, matrices, and determinants. Specifically, you will review properties of the real numbers and complex numbers. You will then learn the Fundamental Theorem of Algebra, which states that every polynomial equation in one variable with complex coefficients has at least one complex solution. You will also review how to solve linear systems of equations and perform operations on matrices. The key is to read through all the material below and complete all the exercises in this unit. The goal of this unit is to ensure that you are comfortable with the key matrix algebra concepts related to Euclidean spaces as these concepts will be referred to throughout this course. The skills and techniques you learn working with matrix theory will be generalized later in the course when you work in a more abstract linear algebra setting.
Time Advisory show close
Learning Outcomes show close
 1.1 Preliminaries

1.1.1 Sets and Functions
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Appendix A: The Language of Sets and Functions”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Appendix A: The Language of Sets and Functions” (PDF)Instructions: Please click on the link above, select the “PDF version of book” link, and read Appendix A for a definition of sets and functions as well as to learn about associated vocabulary for these concepts. You will be using this text throughout the course, so it may help to save the PDF to your desktop for easy reference.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Appendix A: The Language of Sets and Functions”

1.1.2 The Number Line, Real Numbers and Ordered Fields
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries” (PDF)
Instructions: Please read Sections 1.1–1.4 in their entirety (pages 11–15). This reading should be a review.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

1.1.3 Complex Numbers
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries” (PDF)
Instructions: Please read Section 1.5 in its entirety (pages 15–19). Pay particular attention to the polar decomposition of complex numbers and the associated geometric interpretation.
Studying this reading should take approximately 30 minutes to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 2: Introduction to the Complex Numbers”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 2: Introduction to the Complex Numbers” (PDF)
Instructions: If you have not already downloaded the text, please click on the link above. Read Chapter 2. Pay particular attention to the polar decomposition of complex numbers and the associated geometric interpretation.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 2”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 2” (PDF)
Instructions: If you have not already downloaded and saved the text, please click on the link above. Complete calculational exercises 1(a, b, c), 2(b, d), and 4(a, b) and the proofwriting exercises 1, 2, 3, 4, 5 (pages 24 and 25).
Completing this activity should take approximately 3.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

1.1.4 The Fundamental Theorem of Algebra
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “The Fundamental Theorem of Algebra and Factoring Polynomials”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “The Fundamental Theorem of Algebra and Factoring Polynomials” (PDF)Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
Instructions: If you have not already downloaded and saved the text, please click on the link above. Read Chapter 3. Here, you will read about the important Fundamental Theorem of Algebra. In particular, note how the theorem is false when considering polynomials in the real number system.
Studying this reading should take approximately 1 hour to complete.
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 3”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 3” (PDF)
Instructions: If you have not already downloaded and saved the document, please click on the link above. Complete calculational exercise 2 and the proofwriting exercise 3 (pages 34 and 35).
Completing this activity should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “The Fundamental Theorem of Algebra and Factoring Polynomials”

1.1.5 Properties of the Real Numbers
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries” (PDF)
Instructions: Please read Sections 1.7–1.9 (pages 20–26) in their entirety. Here, you will learn some analytic properties that uniquely characterize the real number system.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

1.1.6 Systems of Equations
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries” (PDF)
Instructions: Please read Sections 1.10–1.13 (pages 26–33) in their entirety. Most of this material should be a review.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 1: Preliminaries”

1.2 What Is Linear Algebra?
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “What Is Linear Algebra?”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling's Linear Algebra: As an Introduction to Abstract Mathematics: “What Is Linear Algebra?” (PDF)
Instructions: Please click on the link above, and then read Section 1.2 titled “What Is Linear Algebra?”
Studying this reading should take approximately 30 minutes to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 1”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 1” (PDF)
Instructions: If you have not already downloaded and saved the document to your desktop, please click on the link above and select the “PDF version of book” link. Complete exercises 1b and 2 on page 9 and proofwriting exercise 1 on page 10.
Completing this activity should take approximately 3 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “What Is Linear Algebra?”
 1.3 Matrices and Linear Transformations

1.3.1 Matrices
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems” (PDF)
Instructions: If you have not already downloaded and saved the document to your desktop, please click on the link above. Read Sections 12.1–12.3 in their entirety. Most of this material should be a review.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)
Instructions: Please read Section 2.1 (pages 37–51) in its entirety. Here, you will review some basic properties of matrix algebra.
Studying this reading should take approximately 30 minutes to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)
Instructions: Please work through the oddnumbered problems for 11–29 in Section 2.2 of the textbook on pages 52 and 53. When you are done, check your solutions with the answers on page 487.
Completing this activity should take approximately 4 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems”

1.3.2 Maps and Spaces
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)
Instructions: Please read Sections 2.3–2.6 (pages 53–71) in their entirety. Pay particular attention to the relationship between linear transformations and matrices.
Studying this reading should take approximately 45 minutes to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)
Instructions: Please work through the oddnumbered problems for 1–25 in Section 2.7 on pages 71–73. When you are done, check your solutions with the answers on page 488.
Completing this activity should take approximately 4 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Supplementary Notes on Matrices and Linear Systems” (PDF)
Instructions: If you have not already downloaded and saved the PDF document, click on the link above and select the “PDF version of book” link. Read Sections 12.4 and 12.5 in their entirety. Note the relationship between matrices and linear transformations.
Studying this reading should take approximately 45 minutes to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”

1.4 Vectors and Matrices
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants” (PDF)
Instructions: Please read Chapter 3 (pages 77–104) in its entirety. The determinant of a matrix is an extremely important number associated to the matrix as it provides us with a lot of information about the associated linear transformation.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants” (PDF)
Instructions: Please click on the link above, and work through problems 2, 3, 6, 9, 11, 13, and 15 in Section 3.2 (pages 82 and 83) and problems 5, 6, 8, 9, and 11 in Section 3.6 (pages 102 and 103). When you are done, check your solutions with the answers on page 489.
Completing this activity should take approximately 5 hours.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 3: Determinants”

Unit 2: Spectral Theory
In this unit, you will study Spectral Theory, which refers to the study of eigenvalues and eigenvectors of a matrix. The name Spectral Theory is due to David Hilbert, who coined the phrase in his study of Hilbert space theory. Hilbert's original work was in the setting of quadratic forms, and only later was it discovered that Spectral Theory had applications to quantum mechanics, where it could be used to describe the behavior of atomic spectra. Eigenvalues and eigenvectors of a linear operator are two of the most important concepts in Linear Algebra with applications to many fields, such as computer science (Google's PageRank algorithm), physics (quantum mechanics, vibration analysis), and economics (equilibrium states of Markov models). You will then learn about trace and determinants, two important numbers associated to a matrix. There are several operations that can be applied to a square matrix, and the determinant is a very important operation of this type. The determinant is a number that is calculated from a square matrix and is used to check for many different properties of that matrix, including invertibility. We will learn to compute the determinant and study properties of determinants and the effects of row operations on them. The trace of a matrix is related to the characteristic polynomial of the matrix and can be used to detect nilpotency. You will then learn about Schur's Theorem, which describes how every matrix is related to an upper triangular matrix. Finally, you will learn about quadratic forms, the second derivative test, and some advanced theorems.
Time Advisory show close
Learning Outcomes show close
 2.1 Eigenvalues, Eigenvectors, and Applications

2.1.1 Eigenvalues and Eigenvectors
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors” (PDF)
Instructions: Please read Sections 7.1–7.5. The eigenvectors and eigenvalues of a matrix help to describe the behavior of the associated linear transformation.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please read Section 7.1 (pages 157–164) in its entirety. Work through the eigenvalue/eigenvector examples on your own and check your work with that in the text. Being able to accurately and efficiently perform these computations is essential.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors”

2.1.2 Eigenvalues and Eigenvectors
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please read Section 7.2 (pages 164–167) in its entirety. Here, you will learn about applications of eigenvalues and eigenvectors.
Studying this reading should take approximately 45 minutes to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please work through the oddnumbered problems for 19–33 in Section 7.3 on pages 168–170. When you are done, check your solutions with the answers on page 492.
Completing this activity should take approximately 2.5 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 7: Eigenvalues and Eigenvectors” (PDF)
Instructions: Please read Section 7.6. Note the relationship between the eigenvectors for a rotation matrix and the angle of rotation.
Studying this reading should take approximately 45 minutes to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 7”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 7” (PDF)
Instructions: Please complete calculational exercises 3 and 7 and proofwriting exercises 10, 11, and 12 (pages 95, 96, and 98).
Completing this activity should take approximately 2 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”

2.2 Schur’s Theorem
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please read Section 7.4 (pages 173–180) in its entirety. Schur's Theorem relates any matrix to an associated upper triangular matrix in which the eigenvalues for the original matrix appear on the diagonal. Read through the proof of this theorem and the accompanying lemmas and corollaries.
Studying this reading should take approximately 2 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”

2.3 Trace and Determinant
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please read Section 7.5 (page 180) in its entirety. Pay particular attention to the algebraic properties of the trace.
Studying this reading should take approximately 30 minutes to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 8: Permutations and the Determinant of a Square Matrix”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 8: Permutations and the Determinant of a Square Matrix” (PDF)
Instructions: Please read Chapter 8 in its entirety. Pay particular attention to the algebraic properties of the determinant.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 8”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 8” (PDF)
Instructions: Please complete calculational exercises 1, 4, and 5 and the proofwriting exercises 1, 2, and 3 (pages 115 and 116).
This activity should take approximately 2 hours complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”

2.4 Quadratic Forms, Second Derivative Test, and Advanced Theorems
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please read Sections 7.6–7.9 (pages 181–190) in their entirety. Note how the matrix algebra you have been studying is applied to prove a familiar theorem from multivariable calculus.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory” (PDF)
Instructions: Please work through problems 1, 4, 15, and the oddnumbered problems for 19–31 for Section 7.10. When you are done, check your solutions with the answers on page 493.
This activity should take approximately 3 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 7: Spectral Theory”

Unit 3: Vector Spaces, Fields, Linear Transformations, and Markov Chains
In this unit, we will begin by defining fields and discussing some important examples. Complex numbers (C) will give us insight into some of the key mathematical concepts of linear algebra. We will define vector spaces and study their basic properties before studying finite dimensional vectors spaces. Then, concepts of linear independence, span, bases, subspaces, and dimension are examined in the context of vector spaces. A strong understanding of vector spaces is necessary, because linear algebra is the study of linear maps on vectors spaces. Without a good grasp of vector spaces, understanding linear algebra becomes difficult.
Time Advisory show close
After studying vectors spaces, we will begin to study a special kind of function known as a linear map. These functions arise naturally in linear algebra. We will study linear maps from one vector space to another, as well as linear maps from a vector space to itself (these maps are known as operators and are extremely important in linear algebra). Keep in mind that most of the results in this unit are for finitedimensional vector spaces only. We will then learn how some matrices can be transformed into Jordan canonical form, an upper triangular form for the matrix in which the eigenvalues appear on the diagonal. Note that linear maps have many applications in fields outside of mathematics, such as physics (quantum mechanics, etc.) and engineering (traffic flow, difference equations, etc.). You will finally learn about a certain kind of matrices, called Markov matrices, and see that the existence of the Jordan form is the basis for the proof of limit theorems for Markov matrices.
Learning Outcomes show close
 3.1 Vector Spaces and Fields

3.1.1 Vector Spaces and Their Properties
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields” (PDF)
Instructions: Please read Section 8.1 (pages 199 and 200) in its entirety. The definition of a vector space is an important one, and you should compare the axioms with their more familiar analogs from algebra.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 4: Vector Spaces”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 4: Vector Spaces” (PDF)
Instructions: Pleaseread Sections 4.1 and 4.2. Here the notion of a vector is abstracted. Pay particular attention to the example of polynomial functions, which is the first example of a vector space which is not simply an ordered ntuple of numbers.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”
 3.1.2 Subspaces, Span, and Bases

3.1.2.1 Subspaces
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 4”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 4” (PDF)
Instructions: Please complete calculational exercises 2 and 4 and proofwriting exercises 1, 2, and 3 (pages 46–47).Completing this activity should take approximately 2.5 hours.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields” (PDF)
Instructions: Please read Sections 8.2.1 and 8.2.2 (pages 200–205) in their entirety. The notions of basis and subspace are extremely important ones to master. Read through the examples and proofs on your own. If the proofs are confusing, try working them out in a low dimensional case first.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 4: Vector Spaces”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 4: Vector Spaces” (PDF)
Instructions: Please read Sections 4.3 and 4.4. Work through the examples on your own, and compare your work with that in the text.Studying this reading should take approximately 1 hour to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes, and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 4”

3.1.2.2 Span
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 5: Span and Bases”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 5: Span and Bases” (PDF)Studying this reading should take approximately 1.5 hour to complete.
Instructions: Please read Sections 5.1 and 5.2. The notions of span, basis, independence, and dimension are crucial to understanding linear algebra and its applications.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 5: Span and Bases”

3.1.2.3 Bases
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 5”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 5” (PDF)Completing this activity should take approximately 2.5 hours to complete.
Instructions: Please complete calculational exercises 2 and 5 and proofwriting exercises 2, 4, 5, and 6 (pages 60 and 61).
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields” (PDF)Studying this reading should take approximately 45 minutes to complete.
Instructions: Please read Section 8.2.3 (page 205) in its entirety. Read through the proof of the theorem and make sure you understand why the process presented in the proof must terminate.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 5: Span and Bases”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 5: Span and Bases” (PDF)
Instructions: Please read Sections 5.3 and 5.4. Work through the examples on your own, and compare your work with that in the text.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 5”

3.1.3 Fields
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields” (PDF)
Instructions: Please read Section 8.3 (pages 205–219) in its entirety. Note how the existence of roots of a particular polynomial depends heavily on the field being considered. You should also compare this to the situation regarding the Fundamental Theorem of Algebra.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields” (PDF)
Instructions: Please work through the oddnumbered problems for 1–25 for Section 8.4 on pages 219–221. When you are done, check your solutions with the answers on page 494.
Completing this activity should take approximately 4 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 8: Vector Spaces and Fields”
 3.2 Linear Transformations

3.2.1 Basics
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 6: Linear Maps”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 6: Linear Maps” (PDF)
Instructions: Please read Sections 6.1–6.5 in their entirety. Linear maps are those which preserve the structure of a vector space and are a rich source of additional vector spaces. Work through the examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 2 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 6: Linear Maps”

3.2.2 Matrices and Linear Transformations
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations” (PDF)
Instructions: Please read Sections 9.1–9.3 (pages 225–240) in their entirety. Pay particular attention to the relationship between a linear transformation and an associated matrix.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 6: Linear Maps”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 6: Linear Maps” (PDF)Studying this reading should take approximately 1.5 hours to complete.
Instructions: Please read Sections 6.6 and 6.7 in their entirety. A linear map between two vector spaces can be represented by a matrix, once a basis is chosen for each vector space. Work through the examples on your own, and compare your work with that in the text.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations”

3.2.3 Eigenvalues, Eigenvectors, and Invertibility
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations” (PDF)
Instructions: Please read Section 9.4 (pages 240 and 241) in its entirety. Note the similarities between the spectral theory for matrices and those for more general linear maps.
Studying this reading should take approximately 30 minutes to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations” (PDF)
Instructions: Please click on the link above, and work through problems 1, 7, 11, 13, 15, and 19 in Section 9.5 on pages 242 through 244. When you are done, check your solutions with the answers on page 494.
Completing this activity should take approximately 2.5 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 6”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 6” (PDF)
Instructions: Please complete calculational exercises 1, 3, and 4 and proofwriting exercises 1 and 4 (pages 79 and 80).
Completing this activity should take approximately 2 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 9: Linear Transformations”

3.3 Linear Transformations Canonical Forms
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 10: Linear Transformations Canonical Forms”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 10: Linear Transformations Canonical Forms” (PDF)
Instructions: Please read Chapter 10 (pages 245–274) in its entirety. Pay particular attention to the Jordan canonical form.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 10: Linear Transformations Canonical Forms”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 10: Linear Transformations Canonical Forms” (PDF)
Instructions: Please work through problems 2, 8, 10, 11, and 16 in Section 10.6 (pages 262 and 264) and problems 4–8 in Section 10.9 (pages 273 and 274). When you are done, check your solutions with the answers on pages 494 and 495.
Completing this activity should take approximately 4 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 10: Linear Transformations Canonical Forms”

3.4 Linear Transformations Canonical Forms
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 11: Markov Chains and Migration Processes”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 11: Markov Chains and Migration Processes” (PDF)
Instructions: Please read Chapter 11 (pages 275–286) in its entirety. Markov chains have applications in many fields.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 11: Markov Chains and Migration Processes”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 11: Markov Chains and Migration Processes” (PDF)
Instructions: Please work through problems 1, 6, 8, 9, and 12 in Section 11.4 on pages 284 and 285. When you are done, check your solutions with the answers on page 495.
This activity should take approximately 2.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 11: Markov Chains and Migration Processes”

Unit 4: Inner Product Spaces, Selfadjoint Operators and the Spectral Theorem for Normal Maps
Linear algebra deals with not only Euclidean spaces but also abstract vector spaces. This unit will discuss lengths and angles in an abstract vector space. Inner products allow us to generalize notions such as length, because an inner product is a generalization of a dot product for Euclidean nspace. Having notions of length, angles, and distances in an abstract vector space allow us to apply more tools and methods which help us to better understand the structure of the space.
Time Advisory show close
In this unit, we will discuss inner product spaces, which are vector spaces with an additional structure known as an inner product. Much of the motivation for the subject grew from the need to generalize some geometric properties of twodimensional and threedimensional Euclidean spaces to higher dimensional spaces. In this unit, we will finally begin to understand the geometric aspects of linear algebra, such as representing rotations in the threedimensional Euclidean space as matrices. From this we will understand how to generalize and represent rotations in higher dimensional Euclidean spaces as matrices. The concepts in this unit, such as norm and inner product, provide structure on spaces. We will finally study the basic properties of inner product spaces, orthonormal bases, and the GramSchmidt orthogonalization procedure. We will further study rangenullspace decomposition, orthogonal decomposition, and singularvalue decomposition of spaces.
Next, we will try to understand and answer the question of when a linear operator on an inner product space is diagonalizable. We will study the notion of an adjoint of an operator as well as normal operators and then discuss the spectral theorem, which characterizes the linear operators for which an orthonormal basis consisting of eigenvectors exists. The spectral theory studied here is closely related to that studied in Unit 2. In fact, the eigenvalues and eigenvectors for a matrix are the same as those for the linear transformation determined by the matrix. We will then learn about finding the singularvalue decomposition of an operator. We will conclude by exploring some advanced topics.
Learning Outcomes show close
 4.1 Inner Product Spaces

4.1.1 General Theory
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please read Section 12.1 (pages 287–289) in its entirety. Note that the notion of an inner product space is a generalization of the situation with Euclidean spaces.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces” (PDF)
Instructions: Please read Sections 9.1–9.4. Note the parallels to the situation with Euclidean spaces.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: . These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”

4.1.2 The GramSchmidt Process
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces” (PDF)
Instructions: Please read Section 9.5. The GramSchmidt process can be used to turn any basis for an inner product space into an orthogonal basis.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please read Section 12.2 (pages 289–292) in its entirety. The GramSchmidt process can be used to turn any basis for an inner product space into an orthogonal basis.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”

4.1.3 Advanced Topics
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 9”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 9” (PDF)
Instructions: Please complete calculational exercises 1, 3, and 5 and proofwriting exercises 1, 2, 5, and 6 (pages 133–135).
Completing this activity should take approximately 2.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 9: Inner Product Spaces” (PDF)Studying this reading should take approximately 1 hour to complete.
Instructions: Please read Section 9.6 on pages 128–132. Work through the examples on your own, and compare your work with that in the text.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please read Sections 12.3–12.8 (pages 292–305) in their entirety. If the proofs are difficult to understand, try working through them in the low dimensional setting first.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 12: Inner Product Spaces” (PDF)
Instructions: Please work through problems 1–3, 9, 11, 14, 16, 21, and 22 in Section 12.7 (pages 299–302) and problems 1 and 3 in Section 12.9 (page 306). When you are done, check your solutions with the answers on pages 495 and 496.
This activity should take approximately 4.5 hours to complete.Terms of UseAn Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 9”
 4.2 Spectral Theorem

4.2.1 Self Adjoint Operators
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps” (PDF)
Instructions: Please read Sections 11.1 and 11.2. Work through the examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Sections 13.1 and 13.2 (pages 307–312) in their entirety. Note that Schur's Theorem reappears in this generalized setting.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”

4.2.2 Spectral Theorem
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps” (PDF)
Instructions: Please read Sections 11.3 and 11.4. The Spectral Theorem describes the relationship between normal operators and eigenvectors. Work through the examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Section 13.3 (pages 312–316) in its entirety. Here, you will learn about Hilbert spaces and their properties.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”

4.2.3 Positive and Negative Operators
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Sections 13.4 and 13.5 (pages 317–321) in their entirety. You will read about the possibility of taking fractional powers of certain linear operators.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps” (PDF)
Instructions: Please read Section 11.5. Work through the examples on your own, and compare your work with that in the text.
Studying this reading should take approximately 1.5 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.
 Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”

4.2.4 Decomposition and Applications
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”Links: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps” (PDF)Studying this reading should take approximately 1 hour to complete.
Instructions: Please read Sections 11.6 and 11.7. The singularvalue decomposition generalizes the notion of diagonalization.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Activity: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 11”Link: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Exercises for Chapter 11” (PDF)
Instructions: Please complete calculational exercises 1 and 6 and proofwriting exercises 3 and 4 (pages 158–160).
Completing this activity should take approximately 3 hours to complete.
Terms of Use: These materials have been reproduced for educational and noncommercial purposes and can be viewed in their original format here. Any reproduction or redistribution for commercial use is strictly prohibited.  Reading: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 13: Self Adjoint Operators” (PDF)
Instructions: Please read Sections 13.6–13.11 (pages 322–334) in their entirety. Here, you will learn about the singularvalue decomposition of a matrix, which has applications in statistics and image analysis.
Studying this reading should take approximately 1 hour to complete.
Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.  Activity: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations”Link: Brigham Young University: Kenneth Kuttler’s An Introduction to Linear Algebra: “Chapter 2: Matrices and Linear Transformations” (PDF)
Instructions: Please work through problems 13, 15, 16, and 19 in Section 13.12. When you are done, check your solutions with the answers on page 496.
This activity should take approximately 2 hours to complete.Terms of Use: An Introduction to Linear Algebra was written by Kenneth Kuttler and was relicensed as CCBY 3.0 as part of the Saylor Foundation’s Open Textbook Challenge.
 Reading: UC Davis: Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling’s Linear Algebra: As an Introduction to Abstract Mathematics: “Chapter 11: The Spectral Theorem for Normal Linear Maps”

Final Exam
 Final Exam: The Saylor Foundation's MA212 Final Exam
Link: The Saylor Foundation's “MA212 Final Exam”
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation's MA212 Final Exam