# Fundamentals of Linear Algebra

###### J. S. Chahal
Publisher:
Chapman & Hall/CRC
Publication Date:
2019
Number of Pages:
227
Format:
Hardcover
Series:
Textbooks in Mathematics
Price:
129.95
ISBN:
9781138590502
Category:
Textbook
[Reviewed by
, on
09/22/2019
]
According to its back cover, "Fundamentals of Linear Algebra is like no other book on the subject. By following a natural and unified approach to the subject it has, in less than 250 pages, achieved a more complete coverage of the subject than books with more than twice as many pages.” The book is indeed brief: it weighs barely more than a pound and it is not hard to imagine a diligent lecturer working through every section of the text in a single semester. But there are many versions of a single-semester linear algebra course; how well does this book serve their needs?

• As the textbook for a ”standard” 200-level linear algebra course serving both math and non-math STEM majors. The author says that this is the audience for which the book was written. A glance at the table of contents reinforces this: the book proceeds through matrix algebra and vector spaces, linear maps and determinants, eigenvectors and diagonalization, inner product spaces and orthogonality, the spectral theorem, and ends with a chapter on various applications. This is what one would expect in a standard course. Unfortunately, the lack of computational detail makes it difficult to imagine actually using this book for the purpose. For example, Section 2.3 covers Gaussian elimination: it states the algorithm, provides four examples, and has seven exercises for the reader (five of which have answers in the back). But Example 1 reduces a system of two equations in two variables over $\mathbb{Z/7Z}$, and Example 4 does the same over $\mathbb{Z/6Z}$ (the issue of zero divisors is not raised). Of the seven exercises, only the first two-part exercise asks the reader to work over $\mathbb{Q}$ and both systems are consistent and have unique solutions: the reader is given no practice with the none/one/infinite trichotomy. The author seems aware of this: in the introduction he states that the ”purpose of the theory is not to prepare the students for homework assignments and texts” (p.xii) and that he has ”refrained from providing redundant numerical exercises for repeated drills, a practice which often obscures their real purpose” (p.xii). Presumably, a student would need to find one of the standard introductory texts to get such practice. This computational deficiency ripples through the book: the author never explains explicitly how to find a basis for a nullspace and thus needs to resort to ad hoc methods to find a basis for a 2-dimensional eigenspace in Chapter 6. For students who have never seen eigenvectors before (virtually all of them in such a course), this presents a significant barrier to understanding.
• As the textbook for a ”transition-to-proofs” 200- or 300-level linear algebra course serving math majors.  While the first chapter covers such standard introductory-proofs topics as set theory and proofs by induction, its treatment is sparse: immediately after defining bijection, the author says that it ”is easy to see that $\mathbb{N}$ and $\mathbb{Q}$ have the same cardinality but $\mathbb{Q}$ and $\mathbb{R}$ don’t” (p.6) with no further explanation given. The exercise solutions in the back of the book are strictly computational in nature. The author makes repeated reference to Book 1 of Euclid’s Elements and urges the reader to work through it; perhaps this could be seen as a prerequisite, though I don’t see how this makes the cardinality question easier.
• As the textbook for an ”advanced” or introductory graduate-level linear algebra course.  There is no treatment of normal forms, first of all. The book’s approach is not module-theoretic; it states without proof that a matrix is diagonalizable iff its minimal polynomial has distinct roots to use as an algorithm for detecting diagonalizability. On the back cover, it is claimed that ”the author defines the dimension of a vector space as its Krull dimension. By doing so, most of the facts about bases when the dimension is finite, are trivial consequences of this definition.” While it is true that the book defines dimension using ascending chains, it then immediately defines linear independence and spanning sets and shows that this definition is equivalent to the usual cardinality of a basis. No further reference is made in the book to ascending chains until the last section on field extensions; a lecturer hoping to use this book as a quick concrete introduction to the Zariski topology will be disappointed. Additionally, most results in the book are claimed to hold over arbitrary fields, but this is sometimes careless. The book uses $F$ to denote an arbitrary field throughout the book – including in the chapter on eigenvectors forgetting that the machinery breaks down over finite fields.  The brief sections on cryptography do not mention the significant difficulties that arise for non-prime $n$ for modules over the ring $\mathbb{Z}/n\mathbb{Z}$.
There may be a linear algebra course for which this book is the right choice. On the back cover, it claims to ”attempt to raise expectations and outcomes,” which is a laudable goal. However, I cannot personally imagine teaching linear algebra from this book under any circumstances. I would recommend Linear Algebra and Its Applications by David C. Lay for a "standard" course.

Steve Balady is a visiting assistant professor at Oberlin College.

Preface

1 Preliminaries

What is Linear Algebra?

Rudimentary Set Theory

Cartesian Products

Relations

Concept of a Function

Composite Functions

Fields of Scalars

Techniques for Proving Theorems

2 Matrix Algebra

Matrix Operations

Geometric Meaning of a Matrix Equation

Systems of Linear Equation

Inverse of a Matrix

The Equation Ax=b

Basic Applications

3 Vector Spaces

The Concept of a Vector Space

Subspaces

The Dimension of a Vector Space

Linear Independence

Application of Knowing dim (V)

Coordinates

Rank of a Matrix

4 Linear Maps

Linear Maps

Properties of Linear Maps

Matrix of a Linear Map

Matrix Algebra and Algebra of Linear Maps

Linear Functionals and Duality

Equivalence and Similarity

Application to Higher Order Differential Equations

5 Determinants

Motivation

Properties of Determinants

Existence and Uniqueness of Determinant

Computational Definition of Determinant

Evaluation of Determinants

6 Diagonalization

Motivation

Eigenvalues and Eigenvectors

Cayley-Hamilton Theorem

7 Inner Product Spaces

Inner Product

Fourier Series

Orthogonal and Orthonormal Sets

Gram-Schmidt Process

Orthogonal Projections on Subspaces

8 Linear Algebra over Complex Numbers

Algebra of Complex Numbers

Diagonalization of Matrices with Complex Eigenvalues

Matrices over Complex Numbers

9 Orthonormal Diagonalization

Motivational Introduction

Matrix Representation of a Quadratic Form

Spectral Decompostion

Constrained Optimization-Extrema of Spectrum

Singular Value Decomposition (SVD)

10 Selected Applications of Linear Algebra

System of First Order Linear Differential Equations

Multivariable Calculus

Special Theory of Relativity

Cryptography

Solving Famous Problems from Greek Geometry