# Linear Algebra

David Cherney, UC Davis

Tom Denton, The Fields Institute and York University

Andrew K. Waldon, UC Davis

Copyright Year: 2016

Publisher: University of California, Davis

Language: English

## Formats Available

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

Treats the standard topics in Linear Algebra, plus Linear Optimization (Simplex Method and applications), and the SVD. An exploratory introductory first chapter is non-standard but interesting for engaging students right away in seeing and asking... read more

Treats the standard topics in Linear Algebra, plus Linear Optimization (Simplex Method and applications), and the SVD. An exploratory introductory first chapter is non-standard but interesting for engaging students right away in seeing and asking about the meaning of Linear Algebra. The Index is adequate and has links to pages cited. There was no collected glossary. Cross product is assumed, but then stated a few pages later, and not covered (not untypical for LA books).

Apart from a number of typo's, the mathematical accuracy looks high.

The material covers the standard topics, but the exercise sets and some terms, notations can be nonstandard. The treatment of hyperplanes, and exercises about them were hard for students (and the other faculty they sought help from). The matrix and vector entry superscript-subscript notation is neat for preparing engineers for tensor analysis notations, but is non-standard and somewhat confusing for students at the introductory level.

Longevity depends on the direction the LA curriculum for engineers evolves.

The treatment of hyperplanes, and exercises about them were hard for students (and the other faculty they sought help from). The matrix and vector entry superscript-subscript notation is neat for preparing engineers for tensor analysis notations, but is non-standard and somewhat confusing for students at the introductory level.

Longevity depends on the direction the LA curriculum for engineers evolves.

The lack of clarity to other faculty and to students could be an artifact of the non-standard exposition, problems and notations. Faculty just jump in to help, looking for notations they know. When they have to read through the text, and the students see faculty "do not know how to do the problems", this gives the limping ones an excuse or scandal.

That said, this very quality can cause the more capable and motivated students to learn more, and make cheating or mimicry harder for students to carry off.

This is hard to tell for sure, without a very careful and thoughtful read and use. But, my impression is that the four author effort shows somewhat in a lack of pedagogical consistency.

This is difficult. I would have to skip some sections and chapters in my course, but the non-standard exposition and problems assume the perspectives and exposition of former chapters.

It seems a bit odd or non-standard that Matrices come after vector spaces and after linear transformations. But I think this works in the spirit of the text, which seems to be to introduce the topics after motivation and applied context has been developed.

Text interface is of high quality.

The writing, apart from some typos, is good.

The text is culturally relevant and respectful.

Webworks and Movie ancilarries are supplied!

This text covers the material expected in a first term course on undergraduate Linear Algebra, especially in the considerations of a course with many engineering majors. Major focus is on solving systems of linear equations, Gaussian elimination,... read more

This text covers the material expected in a first term course on undergraduate Linear Algebra, especially in the considerations of a course with many engineering majors. Major focus is on solving systems of linear equations, Gaussian elimination, matrix decompositions, e.g. LU, LUP, bases, determinants, and eigen theory. Several less standard topics are included in the closing chapters including kernels, the simplex algorithm, and idempotent projection operators. Most proofs are held to a minimum or postponed until final chapters thus making it less appropriate for theory based courses.

This text shines in the light of applications involving problems of optimization and tools for numerical methods. The authors do not wait to introduce helpful early notions of length and and angles which aids in discussion of geometric significance. The orthogonalization process is routine but convincing, and the simplex algorithm has been elegantly simplified in the second chapter without losing accuracy.

A surprising weakness in this text is the lack of sparse matrices, minimum polynomials, Krylov methods,, and practical general methods to compute characteristic polynomials. Instead all applications concentrate on archaic topics of LU and QR style problems. This makes the text a less applicable to students of computer and data sciences.

For teaching purposes the book also includes a range of sample midterms and a corpus of WebWork enabled online assignments also available for download.

While the text demonstrates a careful editorial hand, the online assignments are largely ``as is''. For use within

a course it is likely necessary to create new assignments using the exercises contained there in. That process is not completely transparent since exercises are given unhelpful title such as "lecture-10" and these seem to have no correlation with the organization within the text. There are also occasional exercises that are corrupted and not useable within the current versions of WebWork. A final point of concern with the online assignments is that many exercises appear to ask for answers in peculiar formats, perhaps to assist in making them easy to check automatically, however this is certainly not always the case. For example in one instance users are asked to compute eigenvalues and eigenvectors, but the assignment will only accept eigenvectors of unit length.

On the other hand, these reflect a good initial source of problems and substantial investment which will improve overtime with additional exercises.

In most standard ways this text is consistent with traditional philosophies for teaching Linear Algebra.

It is especially sharp about providing simple discussions of matrix decompositions without losing nuances.

However, some unfortunate logic issues arise. For example, the authors define all eigenvectors to be non-zero, but then remark that the set of eigenvectors of a fixed eigenvalue form a subspace ignoring the clear omission they have made of the zero vector. This can be remedied in lectures but it would seem that a text on the subject should settle such confusion preemptively.

An unfortunate illusion occurs in the development of characteristic polynomials. The authors make a wonderful presentation of the determinant and eigenvalues including discussion in geometric terms. They then derive a formula for the determinant polynomial and show how to evaluate it efficiently using elementary matrix operations. However, they do not disclose the impossibility to use such methods to compute the characteristic polynomial (whose entries are variables). Given emphasis on applications it seems surprising not see mention of actual efficient methods to compute characteristic and minimum polynomials. (A single exercise considers this point but the message is lost.)

This book is in the upper range of texts. Compared to alternative text the graphics could be characterized as dated and at times comical. However the content is easily gleaned from the graphics so it does not provide an obstacle to learning. This text will remain in use provided the supplemental material, i.e. online assignments, and the graphics continue to improve.

This is very well written. It reflects multiple perspective: geometric, algebraic, and heuristic. It shows a deep understanding of the topics and a comfort level with teaching. It can be understood by students and taught from easily by first time teachers such as graduate students and post-docs. The only imperative is that each instructor spend adequate time considering the flow of the chapters and sections since some topics are briefly introduced almost as tangents and their complete treatment awaits later development. Skimming the chapter and lure the unprepared instructor into spending too much time on a side topic.

The notation is standard and fitting with the diversity one sees in most of algebra. For example lower case letters are reserved for vectors and numbers, uppercase for matrices and spaces. These conventions are maintained throughout.

Each chapter is self-contained enough that one can easily assemble a course in a different order than the table of contents. In fact the authors include a plausible schedule in their introduction which demonstrates such as permutation of content. On the other hand, there are certain dependences that must be maintained such as presenting determinants before eigentheory. While that dependence is not required by linear algebra, the approach to eigentheory taken in this text relies solely on the characteristic polynomial defined as det(xI-M) and so an other treatment would need to come from supplemental material on Krylov methods.

The text is organized in a familiar manner ideal for those searching first to find applications of linear algebra. It is logic and can be reordered to some degree. It is less flexible for courses electing to focus on theory.

This text is easy to find, download, copy, and print. The PDF offers links that seem not to work but there are instructions on how to modify this to individual courses using the WebWork system.

I have not found any errors, but then if I did it would be the pot calling the kettle black!

This is not culturally relevant, but that seems a criteria I would doubt matters for text in mathematics.

## Table of Contents

- 1 What is Linear Algebra?
- 2 Systems of Linear Equations
- 3 The Simplex Method
- 4 Vectors in Space, n-Vectors
- 5 Vector Spaces
- 6 Linear Transformations
- 7 Matrices
- 8 Determinants
- 9 Subspaces and Spanning Sets
- 10 Linear Independence
- 11 Basis and Dimension
- 12 Eigenvalues and Eigenvectors
- 13 Diagonalization
- 14 Orthonormal Bases and Complements
- 15 Diagonalizing Symmetric Matrices
- 16 Kernel, Range, Nullity, Rank
- 17 Least squares and Singular Values

## Ancillary Material

## About the Book

We believe the entire book can be taught in twenty five 50-minute lectures to a sophomore audience that has been exposed to a one year calculus course. Vector calculus is useful, but not necessary preparation for this book, which attempts to be self-contained. Key concepts are presented multiple times, throughout the book, often first in a more intuitive setting, and then again in a definition, theorem, proof style later on. We do not aim for students to become agile mathematical proof writers, but we do expect them to be able to show and explain why key results hold. We also often use the review exercises to let students discover key results for themselves; before they are presented again in detail later in the book.

The book has been written such that instructors can reorder the chapters (using the La- TeX source) in any (reasonable) order and still have a consistent text. We hammer the notions of abstract vectors and linear transformations hard and early, while at the same time giving students the basic matrix skills necessary to perform computations. Gaussian elimination is followed directly by an “exploration chapter” on the simplex algorithm to open students minds to problems beyond standard linear systems ones. Vectors in Rn and general vector spaces are presented back to back so that students are not stranded with the idea that vectors are just ordered lists of numbers. To this end, we also labor the notion of all functions from a set to the real numbers. In the same vein linear transformations and matrices are presented hand in hand. Once students see that a linear map is specified by its action on a limited set of inputs, they can already understand what a basis is. All the while students are studying linear systems and their solution sets, so after matrices determinants are introduced. This material can proceed rapidly since elementary matrices were already introduced with Gaussian elimination. Only then is a careful discussion of spans, linear independence and dimension given to ready students for a thorough treatment of eigenvectors and diagonalization. The dimension formula therefore appears quite late, since we prefer not to elevate rote computations of column and row spaces to a pedestal. The book ends with applications–least squares and singular values. These are a fun way to end any lecture course. It would also be quite easy to spend any extra time on systems of differential equations and simple Fourier transform problems.

## About the Contributors

### Authors

**David Cherney,** Lecturer, Mathematics, UC Davis. Ph.D., 2010, University of California, Davis.

**Tom Denton. **York University and the Fields Institute, Toronto, Canada. Postdoctoral research with Nantel Bergeron and Mike Zabrocki on k-Schur functions and other topics in algebraic combinatorics. Fulbright Scholar, Maseno University, Kenya. Project concerned using e-learning platforms and emerging technologies to improve the teaching of mathematics in the developing world. Led math camps for secondary students, and co-founded a new technology hub in Kisumu, Kenya. PhD, University of California, Davis.

**Andrew Waldron. **Professor, Mathematics, UC Davis. Waldron's research is devoted to a broad range of problems in theoretical and mathematical physics. In particular he has made an important contribution to the conjectured Banks-Fishler-Shenker-Susskind (BFSS) matrix model of string theory and M-theory. String theory is a proposed perturbative theory that would unify all of the fundamental forces of physics; in other words it is a "theory of everything". Its non-perturbative counterpart is a hypothetical theory called M-theory.