Linear Algebra Done Right [pdf](linear.axler.net)
linear.axler.net
Linear Algebra Done Right [pdf]
https://linear.axler.net/LADR4e.pdf
43 コメント
3B1Bs Essence of Linear Algebra is great, whatever this pdf is is not
thanks for this mate
Why do so many people use the word intuition when they mean understanding?
Because I still don't pretend to fully understand linear algebra. But it's not a mystification anymore, thus "intuition".
"Intuition is the ability to acquire knowledge, without recourse to conscious reasoning or needing an explanation." https://en.wikipedia.org/wiki/Intuition
I don't think this is what you meant to say - have a partial understanding of something is not the same thing as intuition which is more like a "gut feeling".
I don't think this is what you meant to say - have a partial understanding of something is not the same thing as intuition which is more like a "gut feeling".
Intuition is a vehicle for understanding. "Determinant says how much the matrix stretches things if treated like a function" is an intuition but doesn't give full understanding of the determinant.
It's an ancient and likely irrevocable error, like "font" instead of "typeface" (no really, go look it up, we have all been saying "font" when the thing we're talking about is (well, was) "typeface". I only know this because Woz used to tilt against this particular windmill.)
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.
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.
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.
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.
Surprisingly one of the best summaries (~10 pages) to applied linear algebra I've found is in Nielsen and Chuang's Quantum Computation and Quantum Information.
Presented primarily without proofs which whilst argubably can be limiting isn't relevant, at least for what their goal is.
Presented primarily without proofs which whilst argubably can be limiting isn't relevant, at least for what their goal is.
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The same thing applies to computer science. Try figuring out even the basics, like merge or quick sort, using pseudocode in a traditional algorithms book. It's an extremely difficult and time consuming nightmare. But watch a video of how merge or quick sort works, then you gain geniune understanding within minutes.
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.
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.
The homogeneous thing is very nice. As you know it let's you pack translations and self linear maps in K^n into the linear maps of K^(n+1). OpenGL uses it. Also if your perspective is isometric (the projection of the 3D canonic base reminds a Mercedes-Benz logo) it is just linear K^3->K^2.
> 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 cours
I don't think this is the best way to learn about linear algebra. It's still mystifying.
I don't think this is the best way to learn about linear algebra. It's still mystifying.
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
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.
In my experience a more appropriate title for this book would be: 'Linear algebra done ok if this is your second time doing it'. I have seen way too many students who, after having taken a course that used this textbook, could not give an example of a linear operator (yes, I literally asked, show me an example of a linear operator in R^3) because they literally do not have the language for it (because 'matrices are bad').
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)
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)
Somewhat droll isn't it? Everyone else is moving on to topology, and you're specializing in Linear Algebra (done right).
The title always struck me as arrogant, especially since hearing a lot of criticism of its exposition
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."
"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."
I had to look it up:
“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."
“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."
Cool! That is probably the joke that is being made in the book. And it is funny.
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.
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.
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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.
Linear Algebra Done Wrong:
https://www.math.brown.edu/streil/papers/LADW/LADW.html
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doing 2nd year LA right now (as a mature age) have done calc before but this is probably the hardest maths ive been exposed to so far. so many new and abstract concepts to get my head around and try to visualise. and sometimes you just cant visualise it, its abstract. you just have to trust the development of an idea and go "yeah, it generalises i guess"
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
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 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.
I still find Elementary Linear Algebra by Howard Anton more approachable for a beginner. There is a reasion it is in 11th Edition.
I'm guessing this was posted because it's the relatively new 4th edition. Tensors now included!
I'm actually right now working through this book (3rd edition). It's pretty much the most abstract and most clear and simple maths books I've worked through so far (I'm not a mathematician).
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.
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.
well this looks cool , gotta give a try
“The author and his cat Moon”
More authors should include pictures of their pets with their work.
More authors should include pictures of their pets with their work.
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This book is geared to prepare you for functional analysis in a fast pace. It won’t take you so far in applied math, engineering or ML. Considering this, not sure the title is “right”.
You can still find books that use the same approach and covers more.
You can still find books that use the same approach and covers more.
I think the book's name confuses true neophytes who've never even heard of vector space and assume that this book will teach them applied linear algebra. I always thought that this book is for people like me who have already played around with matrices and remember most of the theorems (the intuitive ones), but are left with a creeping sensation of "I know the magic works, but ~why~ does it work?". Needless to say this is a go to reference for me whenever I see a need to refresh something of the basics.
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.
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.
> 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.
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.
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...