omg just had a look and this one is just everything I hate about mathematics and academia.
Starts with lots of random definitions, remarks, axioms and introducing new sign language while completely disregarding introducing what it‘s supposed to do, explain or help with.
All self-aggrandization by creating complexity, zero intuition and simplification. Isn‘t there anybody close to the Feynman of Linear Algebra?
Yeah, a good example is on the second page of the first chapter:
> Remark. It is easy to prove that zero vector 0 is unique, and that given
v ∈ V its additive inverse −v is also unique.
The is the first time the word "unique" is used in the text. Students are going to have no idea whether this is meant in some technical sense or just conventional English. One can imagine various meanings, but that doesn't substitute for real understanding.
This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn.
In terms of intuition building, I think it's probably best to introduce vectors via talking about Euclidean space - which gives the student the possibility of using their physical intuitions. The student should build intuition for how and why vector space "axioms" hold by learning that fundamental operations like addition (which they already grasp) are being extended to vectors in Euclidean space. They already instinctively understand the axiomatic properties being introduced, it's just that the raw technical language being thrown at them fails to connect to any concept they already possess.
A lot of people think Gil Strang was that. Certainly his 18.06SC lecture series is fabulous.[1]
I really like Sheldon Axler and he has made a series of short videos to accompany the book that I think are wonderful. Very clear and easy to understand, but with a little bit more of the intuition behind the proofs etc.
> Isn‘t there anybody close to the Feynman of Linear Algebra?
No. The subject is too young (the first book dedicated to Linear Algebra was written in 1942).
Since then, there have been at least 3 generations of textbooks (the first one was all about matrices and determinants). That was boring. Each subsequent iteration is worse.
What is dual space? What motivates the definition? How useful is the concept? After watching no less than 10 lectures on the subject on youtube, I'm more confused than ever.
Why should I care about different forms of matrix decomposition? What do they buy me? (It turns out, some of them are useful in computer algebra, but the math textbook is mum about it)
My overall impression is: the subject is not well understood. Give it another 100 years. :-)
The thing is, you can teach linear algebra as a gateway to engineering applications or as a gateway to abstract algebra. The second one will require a hell of a lot more conceptual baggage than the first one. It’s also what the book is geared towards.
It is also intended for people who know something about the trade; it isn’t “baby’s first book on maths”. (Why can you graduate high school, do something labelled “maths” for a decade, and still be below the “baby’s first” level, incapable of reading basically any professional text on the subject from the last century? I don’t know. It’s a failure of our society. And I don’t even insist on maths being taught—but if they don’t teach maths, at least they could have the decency to call their stupid two-hundred-year-old zombie something else.)
That conceptual baggage is not useless even in the applied context. For example, I know of no way to explain the Jordan normal form in 19th-century “columns or numbers” style preferred by texts targeted at programmers. (Not point at, not demonstrate, not handwave, explain—make it obvious and inevitable why such a thing must exist.) Or the singular value decomposition, to take a slightly simpler example. (Again, explain. You task, should you choose to accept it, is to see a pretty picture behind it.) And so on.
Again, you can certainly live without understanding any of that. (To some extent. You’ll have a much harder time understanding the motivation behind PageRank then, say. And ordinary differential equations, classical mechanics, or even just multivariable calculus will look much more mysterious than they actually are.) But in that case you need a different book and a different teacher.
I like the free course on linear algebra by Strang’s Ph.D student Pavel Grinfeld. It's a series of short videos with online graded exercises. Most concepts are introduced using geometric vectors, polynomials, and vectors in ℝⁿ as examples. https://www.lem.ma/books/AIApowDnjlDDQrp-uOZVow/landing
> Isn‘t there anybody close to the Feynman of Linear Algebra?
That would probably be Gilbert Strang.
While, as a maths person I would prefer a bit more rigour, his choice of topics and his teaching skill make his the most outstanding introductory course I have seen.
I would run a mile from any course that disrespects determinants. And that includes Axler's!
Also I wish more Linear Algebra courses would cover Generalized Inverses.
As mentioned, the book was intended to be a "second course" in linear algebra. I personally self-studied out of the 3rd edition of Axler, and found it very helpful for understanding exactly what is going on with all the matrix computations we do.
Plus, the same can be said about artists. After all, it's all self-aggrandization, and art is not made to be simple or intuitive.
I actually found the book quite intuitive and helpful in understanding linear algebra. It does explain a lot of the intuition for many definitions, as well as mathematical techniques.
It's easy when presented with new things that you don't understand to reflexively dismiss them, but the ideas here are quite solid. It's also a textbook which aims to introduce students to a slightly higher level of mathematical thinking.
anta40|2 years ago
Guess I need to re-learn it again.
endymi0n|2 years ago
Starts with lots of random definitions, remarks, axioms and introducing new sign language while completely disregarding introducing what it‘s supposed to do, explain or help with.
All self-aggrandization by creating complexity, zero intuition and simplification. Isn‘t there anybody close to the Feynman of Linear Algebra?
bscphil|2 years ago
> Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive inverse −v is also unique.
The is the first time the word "unique" is used in the text. Students are going to have no idea whether this is meant in some technical sense or just conventional English. One can imagine various meanings, but that doesn't substitute for real understanding.
This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn.
In terms of intuition building, I think it's probably best to introduce vectors via talking about Euclidean space - which gives the student the possibility of using their physical intuitions. The student should build intuition for how and why vector space "axioms" hold by learning that fundamental operations like addition (which they already grasp) are being extended to vectors in Euclidean space. They already instinctively understand the axiomatic properties being introduced, it's just that the raw technical language being thrown at them fails to connect to any concept they already possess.
seanhunter|2 years ago
I really like Sheldon Axler and he has made a series of short videos to accompany the book that I think are wonderful. Very clear and easy to understand, but with a little bit more of the intuition behind the proofs etc.
[1] https://youtube.com/playlist?list=PL221E2BBF13BECF6C&si=G2Xq... and https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011...
[2] https://linear.axler.net is his website for the book https://linear.axler.net/LADRvideos.html Is the videos directly although he says the update to the videos to correspond with edition 4 is going to happen 23 Dec.
aidos|2 years ago
https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
resource0x|2 years ago
No. The subject is too young (the first book dedicated to Linear Algebra was written in 1942). Since then, there have been at least 3 generations of textbooks (the first one was all about matrices and determinants). That was boring. Each subsequent iteration is worse.
What is dual space? What motivates the definition? How useful is the concept? After watching no less than 10 lectures on the subject on youtube, I'm more confused than ever.
Why should I care about different forms of matrix decomposition? What do they buy me? (It turns out, some of them are useful in computer algebra, but the math textbook is mum about it)
My overall impression is: the subject is not well understood. Give it another 100 years. :-)
mananaysiempre|2 years ago
It is also intended for people who know something about the trade; it isn’t “baby’s first book on maths”. (Why can you graduate high school, do something labelled “maths” for a decade, and still be below the “baby’s first” level, incapable of reading basically any professional text on the subject from the last century? I don’t know. It’s a failure of our society. And I don’t even insist on maths being taught—but if they don’t teach maths, at least they could have the decency to call their stupid two-hundred-year-old zombie something else.)
That conceptual baggage is not useless even in the applied context. For example, I know of no way to explain the Jordan normal form in 19th-century “columns or numbers” style preferred by texts targeted at programmers. (Not point at, not demonstrate, not handwave, explain—make it obvious and inevitable why such a thing must exist.) Or the singular value decomposition, to take a slightly simpler example. (Again, explain. You task, should you choose to accept it, is to see a pretty picture behind it.) And so on.
Again, you can certainly live without understanding any of that. (To some extent. You’ll have a much harder time understanding the motivation behind PageRank then, say. And ordinary differential equations, classical mechanics, or even just multivariable calculus will look much more mysterious than they actually are.) But in that case you need a different book and a different teacher.
nabla9|2 years ago
Some people have the intuition grasp mathematical concepts more easily than others. Some people don't see it and need to be motivated.
gadrev|2 years ago
Not as big in scope, though, but great introduction.
penguin_booze|2 years ago
He may not be Feynmann, but I'd recommend Pavel's Linear Algebra series: https://www.youtube.com/watch?v=Fnfh8jNqBlg&list=PLlXfTHzgMR.... He does a lot of time developing intuition in the early hours.
bumbledraven|2 years ago
mayd|2 years ago
That would probably be Gilbert Strang.
While, as a maths person I would prefer a bit more rigour, his choice of topics and his teaching skill make his the most outstanding introductory course I have seen.
I would run a mile from any course that disrespects determinants. And that includes Axler's!
Also I wish more Linear Algebra courses would cover Generalized Inverses.
byli|2 years ago
Plus, the same can be said about artists. After all, it's all self-aggrandization, and art is not made to be simple or intuitive.
anon-3988|2 years ago
> It supposed to be a first linear algebra course for mathematically advanced students.
sealeck|2 years ago
It's easy when presented with new things that you don't understand to reflexively dismiss them, but the ideas here are quite solid. It's also a textbook which aims to introduce students to a slightly higher level of mathematical thinking.
danielvaughn|2 years ago
PartiallyTyped|2 years ago
ekm2|2 years ago