# Fundamentals of Matrix Algebra

Gregory Hartman, Virginia Military Institute

Copyright Year: 2011

Publisher: APEX Calculus

Language: English

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## Conditions of Use

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CC BY-NC

## Reviews

The author makes clear in the foreword that this text is not a linear algebra text. It avoids much of the theory associated with linear algebra; although, the author does touch on theorems as necessary. Avoiding theory but using the term... read more

The author makes clear in the foreword that this text is not a linear algebra text. It avoids much of the theory associated with linear algebra; although, the author does touch on theorems as necessary. Avoiding theory but using the term "theorem" might require some discussion in class that is avoided in the textbook. Keeping in mind that this book focuses on computation rather than theory, it covers the main computational aspects of matrix algebra. The section on matrix multiplication has heavy emphasis on square matrices in the examples though the homework uses non-square matrices. This might need supplemented with non-square examples for students to refer to when attempting the homework.

Content-wise, the book seems to be error free. I did not check solutions to all the examples and problems, but the ones I did check were correct.

Linear algebra and matrix algebra doesn't really go out of date. The examples are benign enough not to become outdated. The examples are rather uninteresting. At points the author makes effort to say that the ideas in this book are useful in real life, but the examples are artificial.

There is a quick rush through Reduced Row Echelon Form. After one section, the author assumes the reader is an expert on the topic. I have found this topic can take some students weeks, even months to master. The lack of detail in showing the steps in later sections saves space in the text, but can cause confusion for students. The section on matrix multiplication is a little clunky. The author is trying to avoid the theoretical aspects of a traditional linear algebra course. This leads to questionable notation when introducing matrix multiplication. In essence, the author defines the dot product without using that notation. In the same section, the author multiplies vectors by concatenation (xy means x times y). Because there are multiple ways to multiply vectors, the lack of a sign is ambiguous. However, treating vectors as matrices and there is a standard matrix multiplication for matrices, it would make sense. The author also claims that component-wise matrix multiplication is wrong. While it is not the standard way to multiply matrices, situations arise in which it is the required way. Vector operations are discussed in the chapter on matrix operations. This is the difficulty of the nature of vectors in linear algebra. They are matrices. They are geometric objects. Discussion of on aspect almost requires discussion of the other aspect. However, there is not a "clean" way to do this. The emphasis is on the geometry, without reinforcing the algebraic ideas from the matrix operations sections and then it switches to focusing on the algebra and ignoring the geometry (until a later chapter).

Notation, vocabulary, and such seems consistent throughout. In general, theorems are presented without proof, although in a few sections attempts are proofs are given (perhaps even formal proofs without using that language).

Chapters seem to be rather modular, even if sequential. Most sections, though, end with "guiding questions" for the next section (for example, the section on matrix multiplication ends with questions that infer the matrix inverse will exist, which is explained in the next section). This could cause problems if some sections are skipped, as students are primed for the next section. This does not pose a problem as long as full chapters are used. Each section is appropriate, but begs the next section.

A common problem with texts in linear algebra, which this book faces, is whether to consider vectors or matrices, or both. This book switches back and forth. While there seems to be no good way to handle this, and this book takes the standard (traditional) approach, switching this way can be confusing for students.

The book is a PDF with bookmarks for chapters and sections. All images are clear and very well done.

There are a few minor typos, none that distract from the text (for example, "recieve" instead of "receive"). In other places, spacing is odd.

There is a comment in a footnote about girl and boy names, commenting that a boy has a girl name. It is not necessarily offensive, but it adds nothing to the text. Otherwise, the book is fine. The examples could be more multicultural, but they are generally culturally agnostic.

Overall, the book does what it sets out to do. It teaches matrix algebra with minimal theory and emphasis on computation. It is not completely devoid of theory, and enters the world of proof gently.

## Table of Contents

1 Systems of Linear Equations

- 1.1 Introduction to Linear Equations
- 1.2 Using Matrices To Solve Systems of Linear Equations
- 1.3 Elementary Row Operations and Gaussian Elimination
- 1.4 Existence and Uniqueness of Solutions
- 1.5 Applications of Linear Systems

2 Matrix Arithmetic

- 2.1 Matrix Addition and Scalar Multiplication
- 2.2 Matrix Multiplication
- 2.3 Visualizing Matrix Arithmetic in 2D
- 2.4 Vector Solutions to Linear Systems
- 2.5 Solving Matrix Equations AX = B
- 2.6 The Matrix Inverse
- 2.7 Properties of the Matrix Inverse

3 Operations on Matrices

- 3.1 The Matrix Transpose
- 3.2 The Matrix Trace
- 3.3 The Determinant
- 3.4 Properties of the Determinant
- 3.5 Cramer’s Rule

4 Eigenvalues and Eigenvectors

- 4.1 Eigenvalues and Eigenvectors
- 4.2 Properties of Eigenvalues and Eigenvectors

5 Graphical Explorations of Vectors

- 5.1 Transformations of the Cartesian Plane
- 5.2 Properties of Linear Transformations
- 5.3 Visualizing Vectors: Vectors in Three Dimensions

## About the Book

A college (or advanced high school) level text dealing with the basic principles of matrix and linear algebra. It covers solving systems of linear equations, matrix arithmetic, the determinant, eigenvalues, and linear transformations. Numerous examples are given within the easy to read text. This third edition corrects several errors in the text and updates the font faces.

## About the Contributors

### Author

**Gregory Hartman**, Ph.D., Department Applied Mathematics, Virginia Military Institute