Watch it and if you are a teacher, don't be a smug "formalist".
[1] https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...
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.
I don't think this is the best way to learn about linear algebra. It's still mystifying.
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.
Presented primarily without proofs which whilst argubably can be limiting isn't relevant, at least for what their goal is.
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
Jef Raskin pointed out years ago in his "Humane Interface" that in UI when we say "intuitive" we really mean just "familiar".
His example was the computer mouse. He gave one to an architect (buildings not software) friend and they turned it upside down and used it like a little trackball with their fingertip. (Raskin is that old mice were new.) Few read or heed Raskin.
Personally I find it suits my needs perfectly, even though initially it can be intimidating. But once you start getting the hang of it I think it can allow you to build a much deeper intuition for things than a more applied text.
Or in short - the Yin and Yang of mathematics - 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.
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)
like i always thought of inner products as just dot products but its a whole theory on its own, and cayley hamilton theorm, just been exposed to that but no idea how its useful yet
I still find Elementary Linear Algebra by Howard Anton more approachable for a beginner. There is a reasion it is in 11th Edition.
You can still find books that use the same approach and covers more.
More authors should include pictures of their pets with their work.
"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."
“Be careful not to confuse tortuous with torturous. These two words are relatives—both ultimately come from the Latin verb torquere, which means "to twist," "to wind," or "to wrench"—but tortuous means "winding" or "crooked," whereas torturous means "painfully unpleasant."
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.
