Linear Algebra Done Right
textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
Stats for nerds
130recommend
237mentions
179people
2008–2026years
72% recommend25% neutral3% critical
Mentions per year
What Hacker News says
- Start with 3Blue1Brown's "Essence of Linear Algebra" series https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
Axler's "Linear Algebra Done Right" https://linear.axler.net/LADR4e.pdf
Nielsen and Chuang's "Quantum Computation and Information" Chapter 2. (Available online, but legality depends on where you live)
If you're interested primarily in QM rather than linear algebra, I'd recommend not working through all of Axler, but instead using Nielsen and Chuang to learn about qubits and observables, and using Axler to fill in the gaps.
- The popularity of Linear Algebra Done Right (IMO) comes from two sources:
1. Excellent exercises. Challenging. Really make you put the concepts together.
2. Good, opinionated pedagogy. If you agree with the philosophy (among other things, determinants are not a beginner tool), the explanations are good.
LADR is hardly the only book to eschew determinants for a long time. IIRC Lang takes a similar approach, but is not as digestible.
- Axler’s _Linear Algebra Done Right_ should get you at least part of the way there, content-wise, depending on what other math background you have.
As for reading math papers in general, it’s mostly a process of stepping through it incrementally and trying to verify the steps you don’t understand based on the surrounding context. Most of the concepts in this paper are accessible on Wikipedia or elsewhere, you can make small (e.g. 2 x 2) examples as you go and see what happens.
It’s not an easy skill to acquire from scratch, especially from outside the ivory tower.
- Chapter 7 of Linear Algebra Done Right by Sheldon Axler reads almost as poetry.
- The best textbooks I used in uni were either free or extremely cheap (Linear Algebra done right comes to mind, it was 30-40$, and at least the most recent edition is completely free online. I can't remember if the edition that I bought had that option). More than once I had a professor who was in the process of drafting a textbook or had already written one and it was simply given to everyone in their section for free. Paying hundreds of dollars for intro mathematics books that were glorified collections of practice problems just to get access to the online homework was insulting compared to the care put into those texts some professors gave out for free.
I wouldn't expect every professor to write their own, but I think universities should at least work on some sort of in-house solution for the intro text problem that all the instructors could use, especially public ones. It is absurd that most of those courses are structured to gate homework grades behind an expensive purchase of what is usually a sub-par text.
- Math professor here. If you want to learn math, then for the most part I recommend choosing time-tested avenues, using popular materials.
There are two reasons for this:
(1) Popular materials are usually popular for a reason: they reflect an approximate consensus, across a significant fraction of the mathematical community, that their approaches are more-or-less the best.
(2) If you learn the same way everyone else does, you'll have an easier time talking to others and finding materials on the internet.
I know some very innovative books which I highly recommend, for example Visual Group Theory by Nathan Carter:
https://bookstore.ams.org/clrm-32/
But the innovation is pedagogical, in what Carter chooses to emphasize and how he presents everything. At the book's core, Carter agrees with everyone else about what the foundations of group theory are and should be.
Even Sheldon Axler's Linear Algebra Done Right (another excellent book), with its hilariously provocative title, only differs in its choice of emphasis and order of presentation. His choices are quite compatible with everyone else's.
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