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Author here. I like to look over HN during breakfast, so a surprise this morning. I appreciate folks's kind words. Can I slide in the back story? The course

Author here. I like to look over HN during breakfast, so a surprise this morning. I appreciate folks's kind words. Can I slide in the back story? The course I had as an undergrad was very computational. Matrix multiplication is associative because the indices combine in this way and then you distribute the a_i,j's, etc. I could follow it line-by-line but I didn't feel any real understanding. In grad school I learned that it all makes beautiful sense. So when I started teaching, I wanted an approach that conveyed the sense. I adopted a book that is a great book, very clear and clean, but students did not understand it. They were good students, but they just were not ready. They needed more examples, and by-hand calculations, for instance. I resolved to find a presentation that helped bring them to where they would be ready, or at any rate formed part of a deliberate effort by an undergrad program to work on maturity. I didn't find a book that worked for me, so I wrote one. So although there are a lot of examples and a lot of computation, like the text I had as an undergrad, this development keeps trying to direct the attention of students toward the understanding that they get from higher-level ideas. I often see online where someone asks "What is the right book for XXX?" It seems to me that often there is no one best. Often, it depends on the audience. For the students that I see, this approach accomplishes a lot, it seems to me.

zeebeecee 6y ago
At least for me, doing examples and computations is the best way to learn math, and also very important in research. Often an opaque general statement becomes clear after doing a few small examples. In linear algebra, I personally find some of the courses have too few computations, some concepts are best learned by working an hour by hand on some annoying 6 x 6 matrix..
- cgriswald 6y ago
  I like this course by Philip Klein of Brown University for a computational approach: http://cs.brown.edu/courses/cs053/current/index.htm http://cs.brown.edu/courses/cs053/current/index.htm That said, "working an hour by hand on some annoying 6 x 6 matrix" is how I was taught linear algebra. I got a rare B in the course (even though I aced all the tests!) because there was so much busy work I just refused to do it all. I got literally nothing out of the course in terms of understanding (there was neither time to think nor any real direction given) and a year later I couldn't even do the work anymore. I ended up picking up Axler.
  - saeranv 6y ago
    Agreed. I had an additional conflict in that I am a naturally sloppy person, and the worst part of doing matrix calcs by hand is that I'd often make a calculation error due to an inability to read my own handwriting, which would lead to terrible marks again and again. Learning numpy was godsend.
djtriptych 6y ago
Just an idea: You should allow people to pay more than $22 for the hard copy if they want. Probably a much better way to generate donations than a separate tip payment.
- enriquto 6y ago
  This! I would have happily paid 50 EUR for a high-quality hardcover.
afroisalreadyin 6y ago
Wanted to take this chance to thank you for writing such a lucid and at the same time complete textbook, and then going ahead and making it freely available. It's the first maths textbook that I went through on my own, and might stay the only one, because any other text reads convoluted after yours.
- jimhefferon 6y ago
  What a nice comment. I'm glad to hear it helped; you are welcome.
nextos 6y ago
> I adopted a book that is a great book, very clear and clean, but students did not understand it. Was this book Axler perhaps? A great book that mixes concepts and calculations is Shilov.
- jimhefferon 6y ago
  The book has been posted since 1995, so no, not Axler. :-) I do think that matching the approach to the audience is key. I admire Axler, and a number of other current texts, but in addition to considering the mathematical approach, an instructor also needs to consider where the folks in the room currently are.
  - p1esk 6y ago
    Where does your book assume its audience is?
DennisP 6y ago
That sounds amazing and I think it's exactly what I need. Just reading your website didn't really sell it to me. Maybe you should add a "why another LA book" page and just copy that comment to it.
bigdict 6y ago
>In grad school I learned that it all makes beautiful sense. Is there a page you would refer me to in your book if I asked you about intuition for matrix multiplication?
- commandlinefan 6y ago
  I was afraid to ask the same question. I can do it, I can program it, and I can check to see if it's correct, but I can't for the life of me understand why somebody saw fit to describe matrix multiplication the way they did.
  - jordan_curve 6y ago
    Matrix multiplication is the composition of linear maps. It’s sometimes lost in the more computational approach, but we can think of this geometrically. If you know the first matrix rotates the plan by some amount, and the 2nd rotates it in the same direction as well, then the product must be the rotation matrix that rotates by the sum of the original rotations. That’s a very simple example. Since every nonsingular matrix is just a change of basis, you can get a rich geometric understanding for multiplying matrices. Moreover when things get more complicated, we can use invariants like the determinant and the trace to help guide our intuition. I’d highly suggest watching 3Blue1Brown’s videos on Linear Algebra. He won’t get you to understanding everything (you’ll need to sit down and do problems for that) but he will help you see what intuition is out there in a very beautiful way. He makes a very good point that often when we go through the computations without the geometric intuition, we can spend a ton of time crunching numbers to see results that should have been obvious.
- mixedmath 6y ago
  There is a subsection called matrix multiplication that is about this, and which explicitly mentions the proof of associativity both in the "clear" way and in the "slog through indices" way.
- btilly 6y ago
  Page 238 has the key intuition. But to really understand it you will need some build up to understand linear functions. Matrices are simply a way to write down linear functions, and matrix multiplication is function composition. All of the algebraic properties of function composition are therefore true of matrices and vice versa. That intuition also explains why matrices come up so often. For example consider Calculus. The key idea behind the differential calculus is that if y is close to y0 then f(y) is approximately f(y0) + f'(y0) (y - y0). In the multivariable calculus the same thing is true. Except that f'(y0) is a linear function. Which means that when we write it down we have to write down a matrix.
chinabison 6y ago
I too never really grokked some essential points of linear algebra until I (re)learned it from a chapter in Hungerford's Algebra book, part of Springer's yellow GTM series. I was taught from Strang in undergrand, and while I did well in the course, I often found myself blind as to what all these computations meant. I recall that Strang had said in his preface that the subject had been taught too abstractly and the crucial importance missed, which reading Greub e.g. or other treatises of that era may be true.
tralarpa 6y ago
> The course I had as an undergrad was very computational Do you remember whether that was a course specifically targeting CS students? I am asking because the linear algebra course at "my" university was a course for CS and math students and it was (sadly) not computational at all. If I remember correctly (it's quite some years ago), we started with rings and fields, then vector spaces, and matrices kind of naturally followed from that.
- sidr 6y ago
  > started with rings and fields This seems unnecessary for a first class. I would agree with the parent here that the class should help build intuition and theoretical understanding. Presenting a rich theory with full generality is probably not the best way to do that though. There are some beautiful concepts and geometric understanding to be had with the reals. The abstraction to vector spaces defined over arbitrary rings and fields builds naturally once you have that base.
dboreham 6y ago
Perhaps a trivial question, but I'm wondering about the web site typography not being consistent with the book (serif vs sans)? One looks much nicer on the screen to me than the other.
saeranv 6y ago
> So although there are a lot of examples and a lot of computation, like the text I had as an undergrad, this development keeps trying to direct the attention of students toward the understanding that they get from higher-level ideas. I have not read your book, but this to me is one of the commonalities I have found in all successful curriculums for difficult subjects. An effort to tie new material to old material by repeated comparison. The insight that is brought from persistently trying re-examine previous concepts through the lens of an new one, or vice-versa is so satisfying, and illuminating. Nowadays, I try and force myself to make these kinds of connections and comparisons myself whenever I learn something new.
- leetcrew 6y ago
  reminds me of the semester sophomore year where I was concurrently taking discrete math in the math dept., computer organization (prereq for architecture, basically covers binary number representations and logical implementation of adders, muxers, etc.) in the cs dept., and symbolic logic in the philosophy dept. it was a great feeling when I realized a few weeks into the semester that I was essentially taking three versions of the same course but from very different perspectives.
- jacques_chester 6y ago
  Elaboration and reprocessing is a good way to strengthen memories. The best though is effortful recall -- pulling up a memory that you are close to forgetting.

zeebeecee 6y ago

At least for me, doing examples and computations is the best way to learn math, and also very important in research. Often an opaque general statement becomes clear after doing a few small examples. In linear algebra, I personally find some of the courses have too few computations, some concepts are best learned by working an hour by hand on some annoying 6 x 6 matrix..

cgriswald 6y ago
I like this course by Philip Klein of Brown University for a computational approach: http://cs.brown.edu/courses/cs053/current/index.htm http://cs.brown.edu/courses/cs053/current/index.htm That said, "working an hour by hand on some annoying 6 x 6 matrix" is how I was taught linear algebra. I got a rare B in the course (even though I aced all the tests!) because there was so much busy work I just refused to do it all. I got literally nothing out of the course in terms of understanding (there was neither time to think nor any real direction given) and a year later I couldn't even do the work anymore. I ended up picking up Axler.
- saeranv 6y ago
  Agreed. I had an additional conflict in that I am a naturally sloppy person, and the worst part of doing matrix calcs by hand is that I'd often make a calculation error due to an inability to read my own handwriting, which would lead to terrible marks again and again. Learning numpy was godsend.

djtriptych 6y ago

Just an idea: You should allow people to pay more than $22 for the hard copy if they want. Probably a much better way to generate donations than a separate tip payment.

enriquto 6y ago
This! I would have happily paid 50 EUR for a high-quality hardcover.

afroisalreadyin 6y ago

Wanted to take this chance to thank you for writing such a lucid and at the same time complete textbook, and then going ahead and making it freely available. It's the first maths textbook that I went through on my own, and might stay the only one, because any other text reads convoluted after yours.

jimhefferon 6y ago
What a nice comment. I'm glad to hear it helped; you are welcome.

nextos 6y ago

> I adopted a book that is a great book, very clear and clean, but students did not understand it. Was this book Axler perhaps? A great book that mixes concepts and calculations is Shilov.

jimhefferon 6y ago
The book has been posted since 1995, so no, not Axler. :-) I do think that matching the approach to the audience is key. I admire Axler, and a number of other current texts, but in addition to considering the mathematical approach, an instructor also needs to consider where the folks in the room currently are.
- p1esk 6y ago
  Where does your book assume its audience is?

DennisP 6y ago

That sounds amazing and I think it's exactly what I need. Just reading your website didn't really sell it to me. Maybe you should add a "why another LA book" page and just copy that comment to it.

bigdict 6y ago

>In grad school I learned that it all makes beautiful sense. Is there a page you would refer me to in your book if I asked you about intuition for matrix multiplication?

commandlinefan 6y ago
I was afraid to ask the same question. I can do it, I can program it, and I can check to see if it's correct, but I can't for the life of me understand why somebody saw fit to describe matrix multiplication the way they did.
- jordan_curve 6y ago
  Matrix multiplication is the composition of linear maps. It’s sometimes lost in the more computational approach, but we can think of this geometrically. If you know the first matrix rotates the plan by some amount, and the 2nd rotates it in the same direction as well, then the product must be the rotation matrix that rotates by the sum of the original rotations. That’s a very simple example. Since every nonsingular matrix is just a change of basis, you can get a rich geometric understanding for multiplying matrices. Moreover when things get more complicated, we can use invariants like the determinant and the trace to help guide our intuition. I’d highly suggest watching 3Blue1Brown’s videos on Linear Algebra. He won’t get you to understanding everything (you’ll need to sit down and do problems for that) but he will help you see what intuition is out there in a very beautiful way. He makes a very good point that often when we go through the computations without the geometric intuition, we can spend a ton of time crunching numbers to see results that should have been obvious.
mixedmath 6y ago
There is a subsection called matrix multiplication that is about this, and which explicitly mentions the proof of associativity both in the "clear" way and in the "slog through indices" way.
btilly 6y ago
Page 238 has the key intuition. But to really understand it you will need some build up to understand linear functions. Matrices are simply a way to write down linear functions, and matrix multiplication is function composition. All of the algebraic properties of function composition are therefore true of matrices and vice versa. That intuition also explains why matrices come up so often. For example consider Calculus. The key idea behind the differential calculus is that if y is close to y0 then f(y) is approximately f(y0) + f'(y0) (y - y0). In the multivariable calculus the same thing is true. Except that f'(y0) is a linear function. Which means that when we write it down we have to write down a matrix.

chinabison 6y ago

I too never really grokked some essential points of linear algebra until I (re)learned it from a chapter in Hungerford's Algebra book, part of Springer's yellow GTM series. I was taught from Strang in undergrand, and while I did well in the course, I often found myself blind as to what all these computations meant. I recall that Strang had said in his preface that the subject had been taught too abstractly and the crucial importance missed, which reading Greub e.g. or other treatises of that era may be true.

tralarpa 6y ago

> The course I had as an undergrad was very computational Do you remember whether that was a course specifically targeting CS students? I am asking because the linear algebra course at "my" university was a course for CS and math students and it was (sadly) not computational at all. If I remember correctly (it's quite some years ago), we started with rings and fields, then vector spaces, and matrices kind of naturally followed from that.

sidr 6y ago
> started with rings and fields This seems unnecessary for a first class. I would agree with the parent here that the class should help build intuition and theoretical understanding. Presenting a rich theory with full generality is probably not the best way to do that though. There are some beautiful concepts and geometric understanding to be had with the reals. The abstraction to vector spaces defined over arbitrary rings and fields builds naturally once you have that base.

dboreham 6y ago

Perhaps a trivial question, but I'm wondering about the web site typography not being consistent with the book (serif vs sans)? One looks much nicer on the screen to me than the other.

saeranv 6y ago

> So although there are a lot of examples and a lot of computation, like the text I had as an undergrad, this development keeps trying to direct the attention of students toward the understanding that they get from higher-level ideas. I have not read your book, but this to me is one of the commonalities I have found in all successful curriculums for difficult subjects. An effort to tie new material to old material by repeated comparison. The insight that is brought from persistently trying re-examine previous concepts through the lens of an new one, or vice-versa is so satisfying, and illuminating. Nowadays, I try and force myself to make these kinds of connections and comparisons myself whenever I learn something new.

leetcrew 6y ago
reminds me of the semester sophomore year where I was concurrently taking discrete math in the math dept., computer organization (prereq for architecture, basically covers binary number representations and logical implementation of adders, muxers, etc.) in the cs dept., and symbolic logic in the philosophy dept. it was a great feeling when I realized a few weeks into the semester that I was essentially taking three versions of the same course but from very different perspectives.
jacques_chester 6y ago
Elaboration and reprocessing is a good way to strengthen memories. The best though is effortful recall -- pulling up a memory that you are close to forgetting.