Linear Algebra Roadmap: Prereqs, Books & Topics Guide
Table of Contents
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
Advanced education and experience with mathematics
I think Hoffman and Kunze is a bit abstract for most beginners, having been aimed at a junior level math major at MIT.
after reading some introductory book you might possibly gain something from my notes. In the first 3 pages I summarize the entire content of the most advanced parts of most books, the jordan and rational canonical normal forms. it is free of course. i also like shilov very much, but have not really worked through it. a lot of people, including me, like friedberg insel and spence. but they just don't tell you what is behind the results as i try to do, namely they don't mention the main idea for understanding linear transformations, i.e. minimal polynomials.
http://alpha.math.uga.edu/~roy/laprimexp.pdf
what are your thoughts on Kenneth M Hoffman's Linear Algebra textbook?
This surely fits the list: https://www.lem.ma/Made by: http://drexel.edu/coas/faculty-research/faculty-directory/PavelGrinfeld/
“Thank you micromass for the second insight this month that meets me where I’m at (the other was the one about advanced mathematics for high school students)!
As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I’m done (by what micromass says the other books cover), especially since I’m using a fairly “mathematically pure” book to study from that doesn’t give much motivation the whole subject.
Since it didn’t make it on micromass’ list, I’ll go ahead and mention the book I’m using for anyone who is interested. It’s by Shilov, just called “[URL=’http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X’]Linear Algebra[/URL].” It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to “the point” rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.
Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass’ textbook advice over mine (if only this post had come out before I started to get into the book :H).”
Don’t worry, Shilov is an excellent book. You really can’t go wrong with it. I personally wouldn’t recommend it because it does determinants in the beginning, which I find a fairly unintuitive and perhaps too abstract approach. Also, it never really says what a determinant is geometrically (as far as I recall). But if you like it, it’s a nice book.
Thank you micromass for the second insight this month that meets me where I’m at (the other was the one about advanced mathematics for high school students)!
As someone who is currently attempting to self-study linear algebra (albeit very slowly), this post helped me to see what I can expect to know once I’m done (by what micromass says the other books cover), especially since I’m using a fairly “mathematically pure” book to study from that doesn’t give much motivation the whole subject.
Since it didn’t make it on micromass’ list, I’ll go ahead and mention the book I’m using for anyone who is interested. It’s by Shilov, just called “[URL=’http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X’]Linear Algebra[/URL].” It starts from determinants, which I hear is a different approach than most books take. The second chapter is on linear spaces though, so it seems that it still gets to “the point” rather quickly. I find it excellent, and perhaps it should be considered for someone who is looking into the subject.
Although I will point out that I have no prior experience with linear algebra and have looked at no other texts, so please take micromass’ textbook advice over mine (if only this post had come out before I started to get into the book :H).
One thing about the Macdonald book is how surprisingly small it is (204 pages) for the amount of content it seems to cover. This is mainly for 2 reasons: (1) it handles worked exercises in a cool way and (2) he doesn’t devote space to learning what he calls algorithms (e.g. the mechanistic cookbook recipe for row reduction, etc.)
Regarding worked exercises, the trick is he has you do them! Almost every page has a couple of small exercises that relate to the text you just read. They really make you engage with the content as you go in a neat way that I haven’t seen before. Sometimes you’ll want a little scratch pad and a pencil to work it out and other times it’ll be something simple that you can work out in your head like “what happens if you set t = 0 or 1?” and then you have an aha moment as you realize it simplifies to something you’ve seen before. This is quite rewarding as opposed to being given the same information in a paragraph.
Regarding algorithms, an example is matrix inversion – he goes through the concept and applications of it, thereafter using it throughout the book but he does not devote space to building up the detailed recipe for mechanistically computing one by hand. Same goes for row reduction, determinants, eigenstuff, etc. In the Preface he argues that the recipes are not needed for theoretical development, and no one solves them by hand anymore anyways except as exercises in Linear Algebra textbooks.
Anyone intending to tackle both the Linear Algebra Insight and the Intro Analysis Insight, will probably notice that there is some overlap between the two. Micromass was kind enough to provide an efficient way to navigate through them, which he gave permission to repost here:
“So if you’re doing both of them, then I would recommend:
Do Bloch Analysis and MacDonald in parallel.
Then after Bloch do Hubbard, and after MacDonald do Axler.
This way you’ll get everything without too much repetition. MacDonald will teach you the basics of LA (vector spaces, linear transformations), but will also do geometric algebra. Hubbard will repeat the basics but not from a point of view of analysis. And Axler will do things in the most rigorous light. Avoiding determinants in Axler is not a problem since Hubbard and MacDonald cover those. What do you think? It is possible to do Treil instead of Axler if you prefer Treil, but it’s really up to you.”
Great list – thanks for putting this together!
The author of Linear and Geometric Algebra, Alan Macdonald, has put together some youtube videos in support of his book.
[URL]https://www.youtube.com/c/AlanMacdonald1/playlists[/URL]
One thing about the Macdonald book is how surprisingly small it is (204 pages) for the amount of content it seems to cover. This is mainly for 2 reasons: (1) it handles worked exercises in a cool way and (2) he doesn't devote space to learning what he calls algorithms (e.g. the mechanistic cookbook recipe for row reduction, etc.)Regarding worked exercises, the trick is he has you do them! Almost every page has a couple of small exercises that relate to the text you just read. They really make you engage with the content as you go in a neat way that I haven't seen before. Sometimes you'll want a little scratch pad and a pencil to work it out and other times it'll be something simple that you can work out in your head like "what happens if you set t = 0 or 1?" and then you have an aha moment as you realize it simplifies to something you've seen before. This is quite rewarding as opposed to being given the same information in a paragraph.Regarding algorithms, an example is matrix inversion – he goes through the concept and applications of it, thereafter using it throughout the book but he does not devote space to building up the detailed recipe for mechanistically computing one by hand. Same goes for row reduction, determinants, eigenstuff, etc. In the Preface he argues that the recipes are not needed for theoretical development, and no one solves them by hand anymore anyways except as exercises in Linear Algebra textbooks.
Anyone intending to tackle both the Linear Algebra Insight and the Intro Analysis Insight, will probably notice that there is some overlap between the two. Micromass was kind enough to provide an efficient way to navigate through them, which he gave permission to repost here:"So if you're doing both of them, then I would recommend:Do Bloch Analysis and MacDonald in parallel.Then after Bloch do Hubbard, and after MacDonald do Axler.This way you'll get everything without too much repetition. MacDonald will teach you the basics of LA (vector spaces, linear transformations), but will also do geometric algebra. Hubbard will repeat the basics but not from a point of view of analysis. And Axler will do things in the most rigorous light. Avoiding determinants in Axler is not a problem since Hubbard and MacDonald cover those. What do you think? It is possible to do Treil instead of Axler if you prefer Treil, but it's really up to you."
Great list – thanks for putting this together!The author of Linear and Geometric Algebra, Alan Macdonald, has put together some youtube videos in support of his book.https://www.youtube.com/c/AlanMacdonald1/playlists