Linear Algebra Done Right [pdf]

thanks for this mate

Why do so many people use the word intuition when they mean understanding?

For what it’s worth, Axler’s book did help me considerably.

Logged in to comment this. Very much agree. It's crazy (to me) that books like this gets so much hype on HN when there are much better videos that explain these topics better.

Most of my math learning starting in high school all the way through undergrad was done by watching YouTube videos. I used books to practice problems, but when it came to understanding topics more deeply, it was always some random person on YouTube who did it better.

I hope in the future, all math (at least applied math) is explained using nice visualizations + videos instead of books like this.

I think a good project-based way to learn about linear algebra is to build a 3d scene rasterizer from scratch. Once you get nested transforms and projection matrices figured out, you are probably going to have a much deeper sense for what these things actually are than someone walking out of a typical college course. The 4x4 homogenous transformation matrix sort of tricked me into learning about things I didn't originally intend to. I spent a solid week watching videos (such as the exact one you linked above) to reach a more satisfying understanding.

I vividly recall manipulating matrices on paper for weeks in college. Absolutely none of it had any real meaning to me. Very little of that education helped me as I got into actual projects, other than some vague awareness that there are these things called matrices and something about system of equations. Having something to pull me towards a specific objective has always resulted in a much more substantial education. Learning about some of this stuff in isolation is really quite painful. Learning about painful things to get to the fun bits seems more tenable.

Formalism isn't really the problem though. The problem is that it needs to be accompanied with geometrical explanations and intuition (especially for linear algebra).

Formalism is here to help you put words on intuition. Intuition without formalism is as useless as formalism without intuition.

In linear algebra in particular, people who avoid formalism at all cost tend to focus on obscure calculation results on matrices instead on their more geometrical counterparts on linear maps

Different audiences. 3B1B can be very effective for intuition. It does not, however, prepare you for proofs or anything rigorous or thinking about things in an axiomatic way. For example, it isn't going to provide a basis for anything infinite dimensional.

3B1Bs Essence of Linear Algebra is great, whatever this pdf is is not

> sometimes you need the excellent dry theory and sometimes you need the more concrete but messy application, and in truth you will always vacillate between the two - this is the former

Fully agree. Axler says it quite clearly that the book is intended for a second course in linear algebra. While not an applied text, I find it close enough that it allows you to subsequently go back to your applied material and see it with new eyes.

While there may be genuine issues with the book, especially when used as a first text, this strikes me as evidence either that the book was only taught through a few chapters or that the students simply didn't understand anything from it (which may be the fault of the book, of course -- but it also may be the fault of the instructor, or the preparedness of the students for the course).

Chapters 5-8 are all on operators (i.e., the entire second half of the book!). One of the most common exercises in the book is "give an example of..." And chapters 7 and 8 are literally titled "Operators on Inner Product Spaces" and "Operators on Complex Vector Spaces." If you can complete the homework with a passing grade and then pass an exam covering that material, there's no way you don't know examples of operators. Possibly you forgot the definition, but a quick, one-sentence reminder of that should make it easy to list plenty of examples)

I consider myself an expert in linear algebra and I still find Cayley-Hamilton perplexing. Like it's clearly true but the proofs are unsatisfying It feels like there is an algebraic system that would make it obvious but hasn't been discovered yet (or at least I'm not aware of it).

I just glanced through LADR. I am not sure whether this is the best way to approach LA for a beginner.

I still find Elementary Linear Algebra by Howard Anton more approachable for a beginner. There is a reasion it is in 11th Edition.

There is a book ripping on the title listed in the thread. I don't think I see it as arrogant although it's definitely an opinionated title.

What was the substance of the criticism?

There is a meeting or mismeeting between book and reader in math. Sometimes you are on the wrong footing to absorb a book. You bounce. Maybe you come back later and absorb the book.

As far as the title, just catching marketing I think. Might not appeal to all for sure.

For what it's worth:

"The title of this book deserves an explanation. Most linear algebra textbooks use determinants to prove that every linear operator on a finite-dimensional complex vector space has an eigenvalue. Determinants are difficult, nonintuitive, and often defined without motivation. To prove the theorem about existence of eigenvalues on complex vector spaces, most books must define determinants, prove that a linear operator is not invertible if and only if its determinant equals 0, and then define the characteristic polynomial. This tortuous (torturous?) path gives students little feeling for why eigenvalues exist.

In contrast, the simple determinant-free proofs presented here (for example, see 5.19) offer more insight. Once determinants have been moved to the end of the book, a new route opens to the main goal of linear algebra—understanding the structure of linear operators."