Linear Algebra Roadmap: Prereqs, Books & Topics Guide

Overview

This article gives a roadmap for students to learn the basics of linear algebra. Aside from calculus, linear algebra is one of the most applicable subjects in mathematics: it is widely used in engineering, the sciences, and computer science. The best way to approach linear algebra is with a focus on vector spaces and linear transformations, so the recommendations below follow that perspective.

Prerequisites

Prerequisite skills

Besides standard high-school algebra, geometry, and trigonometry (calculus is not a formal prerequisite), there are two specific prerequisites for studying linear algebra:

• You should be familiar with proof techniques.
• You should have seen the basics of matrices and determinants.

Recommended proof books

For learning proofs, I recommend the following books (no particular order):

• Velleman, How to Prove It: A Structured Approach — http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
• Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics — http://www.amazon.com/Proofs-Fundamentals-Abstract-Mathematics-Undergraduate/dp/1441971262
• Hammack, The Book of Proof — http://www.people.vcu.edu/~rhammack/BookOfProof/

These books introduce proof-based mathematics and the basic set-theory notation and assumptions that are necessary for abstract algebra and linear algebra.

Matrix & determinant basics

Basic matrix and determinant material is usually covered in high-school or introductory college texts. Examples include:

• Lang, Basic Mathematics — http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877
• OpenStax, College Algebra — https://openstaxcollege.org/textbooks/college-algebra
• Khan Academy, Linear algebra (matrix transformations) — https://www.khanacademy.org/math/linear-algebra/matrix_transformations

You do not need complete mastery of matrices and determinants before starting linear algebra. Familiarity with 2×2 and 3×3 examples, computing matrix inverses, and solving linear systems is sufficient; the recommended linear algebra texts will develop the full theory and build intuition for matrices and determinants as you study vector spaces.

Introduction to linear algebra

Core emphasis

Proper linear algebra emphasizes vector spaces and linear transformations between them. It rigorously develops the theory of n×n matrices and determinants and applies these concepts in geometric and algebraic contexts.

Recommended introductory text

As an introductory textbook I highly recommend Friedberg, Insel, and Spence, Linear Algebra — http://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514. This modern introduction treats the following topics:

Topics covered

• Vector spaces (including linear dependence, subspaces, bases, dimension)
• Linear transformations and matrices (including rank, isomorphisms, change-of-basis matrices, dual spaces)
• Determinants with rigorous proofs of properties
• Diagonalization (including eigenvalues, Markov chains, and the Cayley–Hamilton theorem)
• Inner product spaces (including the Gram–Schmidt process, the spectral theorem, singular value decomposition)
• Canonical forms (including the Jordan and rational canonical forms)

For an introduction to the world of algebras and related perspectives, I prefer Friedberg, Insel, and Spence over other texts, but there are several other books I like as follow-up or alternative references.

Recommended textbooks and resources

Linear Algebra Done Right — Axler

Axler’s Linear Algebra Done Right contains many elegant proofs and useful perspectives. I do not recommend it as a first introduction because it avoids determinants until a very abstract final chapter. Determinants are important in many applications, so I recommend a more conventional introduction before using Axler as a supplement. Many proofs (for example, existence-of-eigenvector arguments) are particularly instructive.

http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799

Prerequisites: A course on linear algebra such as Friedberg, Insel, and Spence

Linear Algebra Done Wrong — Treil

Treil’s Linear Algebra Done Wrong has long been a favorite. It is terse, so it is better as a second book. It covers nontraditional topics (for example, tensors) and is freely available online.

https://www.math.brown.edu/~treil/papers/LADW/LADW.html

Prerequisites: A course on linear algebra such as Friedberg, Insel, and Spence

Matrix Analysis and Applied Linear Algebra — Meyer

Meyer’s book focuses on the applied, numerical side of linear algebra and contains practical insights and computational rules that are not always found in theoretical texts. For example, misconceptions about determinants and near-singularity are addressed in detail.

http://www.amazon.com/Carl-D-Meyer-Analysis-Applied/dp/B008UB4KJI

Prerequisites: None

3000 Solved Problems in Linear Algebra — Lipschutz

If you need many exercises, this book is filled with computational problems and some proof-based problems.

http://www.amazon.com/000-Solved-Problems-Linear-Algebra/dp/0070380236

Prerequisites: None

Linear and Geometric Algebra — Macdonald

Macdonald’s book covers both linear algebra and geometric algebra (Clifford algebras). Geometric algebra provides a unified viewpoint for quaternions, forms, and related structures and offers an alternative interpretation of the determinant that introductory courses often omit.

http://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932

Prerequisites: None

Geometric Algebra for Computer Science — Dorst, Fontijne, Mann

Despite the “computer science” label, this is a substantial treatment of geometric algebra that provides intuition useful in physics and computer science. It covers more of geometric algebra than Macdonald’s book.

http://www.amazon.com/Geometric-Algebra-Computer-Science-Revised/dp/0123749425

Prerequisites: None

Linear Algebra via Exterior Products — Winitzki

Winitzki presents a basis-free approach that emphasizes the wedge product rather than coordinate computations. It is a good companion for geometric algebra and works well as a sequel to standard linear algebra courses.

http://www.amazon.com/Linear-Algebra-via-Exterior-Products/dp/140929496X?tag=pfamazon01-20

Prerequisites: A course in linear algebra such as Friedberg, Insel, and Spence

Advanced Linear Algebra — Roman

Roman’s Advanced Linear Algebra offers a graduate-level, advanced perspective, covering topics such as modules, Hilbert spaces, and umbral calculus. It requires mathematical maturity: analysis, abstract algebra, and prior linear algebra are necessary.

http://www.amazon.com/Advanced-Linear-Algebra-Graduate-Mathematics/dp/0387728287

Prerequisites: Analysis, abstract algebra, and linear algebra

Micromass

Advanced education and experience with mathematics