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I'm not too fond of the example-driven approach. The student will certainly be able to multiply simple matrices and compute determinants in some simple cases, b

I'm not too fond of the example-driven approach. The student will certainly be able to multiply simple matrices and compute determinants in some simple cases, but will not be able to answer fundamental questions like "what can I put in a matrix?", or "what is a vector field?" (the author gives 10 "rules" with no structure, so it's impossible for the student to memorize them). The author throws around `R` everywhere, but all his examples use `Z`. For a book on linear _algebra_, there is very little algebraic rigor in the book. The standard textbook for learning linear algebra at the undergraduate level is Hoffman and Kunz, which is far better. In short, this is a poor textbook.

jsinai 6y ago
My first course in linear algebra was extremely formal with very little practical examples or visual intuition. It’s easy not to appreciate just how much intuition can go a long way in helping us understand a concept, even if the intuition is formally “fuzzy”. I used Hefferon’s Linear Algebra to plug the gaps in my intuitive and practical knowledge, and was all the better for it. People have different reasons and approaches for learning, and in the linear algebra landscape Hefferon is a worthy addition. (edit: just like one of the commenters below I bought the hard copy a few years ago to show my support and thanks for the free edition.)
solinent 6y ago
In my experience teaching my friends and colleagues there are two ends of the learning spectrum--you can learn by example first or by theory first, or both, but the former is by far the lion's share of the population. I've taught people where there's absolutely no way I can teach them a generalization unless it's rote--however they can get domain-specific knowledge and beat me over the head in their domain and teach me something about my generalization as well that I might have been missing. The best way for me to teach calculus to most people is to walk around a room explaining concepts like velocity, speed, acceleration, and position. However, I personally would never learn it this way--I prefer Spivak, during my read through I learned more than I could ever learn in a class. It's also my belief that most people aren't like me in the sense that they learn from example first. However, most people I know are like me so it's hard to realize how needed a textbook like this is and how huge it could be for the regular person to learn something like this.
- skohan 6y ago
  If you think about it, pretty much everyone learns math through example first. You learn arithmetic in elementary school, and it might only be after an additional decade or more of schooling that you actually learn the logical underpinnings which make the basic arithmetic operations work. If you actually started with boolean logic and tried to build up all of mathematics from there without at some point tying it back to a tangible example which can be understood in terms of the real world I'm not sure anyone could successfully learn it.
  - solinent 6y ago
    So I actually had a basic algebra system before they taught it to me, for what it's worth, probably something I figured out from the structure of the language my parents were using (both accountants). I'm a pretty rare individual, I've never had to work at math. I took a long time to learn how to learn from example, it used to be impossible but now it's almost just as natural.
  - artagnon 6y ago
    True. Examples are necessary for both motivation, and for showing how the formalisms work, at this level. One good undergraduate textbook that has plenty of examples is Dummit & Foote. However, that doesn't mean that you don't define anything properly, and solely use examples to show how to crunch certain computations. The student is then stuck with a bunch of patterns in their heads, with little mathematical understanding.
- deleted 6y ago
  [deleted]
dash2 6y ago
Thanks for your perspective. Not sure why you got the downvotes. What do you think of "Linear Algebra Done Right"?
- strangeattractr 6y ago
  I think it's the leap from - 'I'm not a fan of the approach' to 'this is a poor textbook'. Just because they don't like the approach doesn't mean it won't be effective for other learners. Some people absolutely need examples before they can attach anything theoretical to a concept.
  - amelius 6y ago
    Yes, the wording in that comment is not very friendly. Also, I think that examples motivate people. I'm sure a lot of people are capable of understanding highly abstract things, but motivation is key.
- artagnon 6y ago
  Interesting approach. He starts off with vector spaces, after providing some intuition, and uses it to drive the rest of the book. He defines matrices in terms of linear maps, and talks about operations on vector spaces, with small sections dedicated to the corresponding matrix computations. There's no hand-waving or lack of rigor. Overall, a fine textbook.
- threatofrain 6y ago
  There's also the Terse Introduction to Linear Algebra, which is shorter and more rigorous than Linear Algebra Done Right. All of these are known texts which have generated some body of discussion. An idiosyncrasy of Axler's book is that it introduces determinants only toward the end.
  - nsajko 6y ago
    > An idiosyncrasy of Axler's book is that it introduces determinants only toward the end I think the idea is that determinants do not necessarily belong in linear algebra proper, at least not with as prominent a position as they are usually awarded in textbooks. More specifically the idea is that determinants are being excessively and detrimentally used in linear algebra proofs. Axler actually published a paper called "Down With Determinants!" in 1995, for which he won the Lester R. Ford Award. Quote from the introduction: > This paper focuses on showing that determinants should be banished from much of the theoretical part of linear algebra. Determinants are also useless in the computational part of linear algebra. Paper: https://www.maa.org/sites/default/files/pdf/awards/Axler-Ford-1996.pdf https://www.maa.org/sites/default/files/pdf/awards/Axler-For... Wikipedia page for the award: https://en.wikipedia.org/wiki/Paul_R._Halmos_%E2%80%93_Lester_R._Ford_Award https://en.wikipedia.org/wiki/Paul_R._Halmos_%E2%80%93_Leste...
Tainnor 6y ago
Aren't these the standard vector space axioms? I seem to recall that they were defined in a quite similar way in my class. Of course, you can simply the definition if you can say "vector spaces form a group and also satisfy these axioms with respect to scalar multiplication", but then you need to introduce groups first, which would be a distraction.
WoahNoun 6y ago
How is "what can I put in a matrix?" a fundamental question? Looking at the textbook, the author certainly covers the fundamental questions in vector spaces, linear maps, and decomposition/normal forms. There's even a really great section on projective geometry that would have been helpful before seeing it quickly defined as homogenous coordinates before moving on to the definition as a quotient manifold of a sphere and Lie group. Are you complaining about the axioms for a vector space? Because that list will exist for any LA class lol. That's like complaining that a college Calculus class is teaching the limit definition of the derivative.
- Tainnor 6y ago
  I think (from only having skimmed the contents) that there is some handwaving around the definition of the characteristic polynomial. In order to properly define it, you need to allow matrices over arbitrary rings, not just fields.

jsinai 6y ago

My first course in linear algebra was extremely formal with very little practical examples or visual intuition. It’s easy not to appreciate just how much intuition can go a long way in helping us understand a concept, even if the intuition is formally “fuzzy”. I used Hefferon’s Linear Algebra to plug the gaps in my intuitive and practical knowledge, and was all the better for it. People have different reasons and approaches for learning, and in the linear algebra landscape Hefferon is a worthy addition. (edit: just like one of the commenters below I bought the hard copy a few years ago to show my support and thanks for the free edition.)

solinent 6y ago

In my experience teaching my friends and colleagues there are two ends of the learning spectrum--you can learn by example first or by theory first, or both, but the former is by far the lion's share of the population. I've taught people where there's absolutely no way I can teach them a generalization unless it's rote--however they can get domain-specific knowledge and beat me over the head in their domain and teach me something about my generalization as well that I might have been missing. The best way for me to teach calculus to most people is to walk around a room explaining concepts like velocity, speed, acceleration, and position. However, I personally would never learn it this way--I prefer Spivak, during my read through I learned more than I could ever learn in a class. It's also my belief that most people aren't like me in the sense that they learn from example first. However, most people I know are like me so it's hard to realize how needed a textbook like this is and how huge it could be for the regular person to learn something like this.

skohan 6y ago
If you think about it, pretty much everyone learns math through example first. You learn arithmetic in elementary school, and it might only be after an additional decade or more of schooling that you actually learn the logical underpinnings which make the basic arithmetic operations work. If you actually started with boolean logic and tried to build up all of mathematics from there without at some point tying it back to a tangible example which can be understood in terms of the real world I'm not sure anyone could successfully learn it.
- solinent 6y ago
  So I actually had a basic algebra system before they taught it to me, for what it's worth, probably something I figured out from the structure of the language my parents were using (both accountants). I'm a pretty rare individual, I've never had to work at math. I took a long time to learn how to learn from example, it used to be impossible but now it's almost just as natural.
- artagnon 6y ago
  True. Examples are necessary for both motivation, and for showing how the formalisms work, at this level. One good undergraduate textbook that has plenty of examples is Dummit & Foote. However, that doesn't mean that you don't define anything properly, and solely use examples to show how to crunch certain computations. The student is then stuck with a bunch of patterns in their heads, with little mathematical understanding.
deleted 6y ago
[deleted]

dash2 6y ago

Thanks for your perspective. Not sure why you got the downvotes. What do you think of "Linear Algebra Done Right"?

strangeattractr 6y ago
I think it's the leap from - 'I'm not a fan of the approach' to 'this is a poor textbook'. Just because they don't like the approach doesn't mean it won't be effective for other learners. Some people absolutely need examples before they can attach anything theoretical to a concept.
- amelius 6y ago
  Yes, the wording in that comment is not very friendly. Also, I think that examples motivate people. I'm sure a lot of people are capable of understanding highly abstract things, but motivation is key.
artagnon 6y ago
Interesting approach. He starts off with vector spaces, after providing some intuition, and uses it to drive the rest of the book. He defines matrices in terms of linear maps, and talks about operations on vector spaces, with small sections dedicated to the corresponding matrix computations. There's no hand-waving or lack of rigor. Overall, a fine textbook.
threatofrain 6y ago
There's also the Terse Introduction to Linear Algebra, which is shorter and more rigorous than Linear Algebra Done Right. All of these are known texts which have generated some body of discussion. An idiosyncrasy of Axler's book is that it introduces determinants only toward the end.
- nsajko 6y ago
  > An idiosyncrasy of Axler's book is that it introduces determinants only toward the end I think the idea is that determinants do not necessarily belong in linear algebra proper, at least not with as prominent a position as they are usually awarded in textbooks. More specifically the idea is that determinants are being excessively and detrimentally used in linear algebra proofs. Axler actually published a paper called "Down With Determinants!" in 1995, for which he won the Lester R. Ford Award. Quote from the introduction: > This paper focuses on showing that determinants should be banished from much of the theoretical part of linear algebra. Determinants are also useless in the computational part of linear algebra. Paper: https://www.maa.org/sites/default/files/pdf/awards/Axler-Ford-1996.pdf https://www.maa.org/sites/default/files/pdf/awards/Axler-For... Wikipedia page for the award: https://en.wikipedia.org/wiki/Paul_R._Halmos_%E2%80%93_Lester_R._Ford_Award https://en.wikipedia.org/wiki/Paul_R._Halmos_%E2%80%93_Leste...

Tainnor 6y ago

Aren't these the standard vector space axioms? I seem to recall that they were defined in a quite similar way in my class. Of course, you can simply the definition if you can say "vector spaces form a group and also satisfy these axioms with respect to scalar multiplication", but then you need to introduce groups first, which would be a distraction.

WoahNoun 6y ago

How is "what can I put in a matrix?" a fundamental question? Looking at the textbook, the author certainly covers the fundamental questions in vector spaces, linear maps, and decomposition/normal forms. There's even a really great section on projective geometry that would have been helpful before seeing it quickly defined as homogenous coordinates before moving on to the definition as a quotient manifold of a sphere and Lie group. Are you complaining about the axioms for a vector space? Because that list will exist for any LA class lol. That's like complaining that a college Calculus class is teaching the limit definition of the derivative.

Tainnor 6y ago
I think (from only having skimmed the contents) that there is some handwaving around the definition of the characteristic polynomial. In order to properly define it, you need to allow matrices over arbitrary rings, not just fields.