> 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.

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.

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.

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.

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.

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

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.

> 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.

> 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.

>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?

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.

Add me to the list of people who have the same question, but with a slightly different perspective.

I understand the interpretation of matrix multiplication as a linear combinations, but I would also like to understand what is occurring geometrically. If I matrix multiply a 2D vector <a, b> by the basis vectors <<0, 1>, <1, 0>>, then I understand I am "weighting" an x, and y unit vector (as defined by my basis) by the x-component and y-component of my <a, b> vector, and then adding them together. So geometrically, I'm stretching by a factor of a, and b, then summing to get my resulting vector.

So, the way I'm trying to think of it geometrically is: any matrix is just a representation of some kind of basis (i.e some kind of linear system). I picture this as a sort of orthogonal cartesian grid that we squish and stretch while still being a series of intersecting parallel lines.

Matrix multiplication, is therefore taking in some vector, and "weighting" the basis vectors represented by our matrix by the components of that vector, and then summing them. To put it another way, we're essentially mapping a vector into that new basis system represented by our matrix.

Is that correct? Is there a better way to try and think of matrix multiplication geometrically?

The subsection on matrix multiplication is on p 228. But, of course, a person can't just dip in, they need to be in the flow of the prior 227 pages.

There are two relevant facts:

    1. Linear maps are determined by their behavior on basis elements
    2. The composition of linear maps is still linear

Together, this means you can work out how the composition of two linear maps affects the coefficients of each vector (relative to a fixed basis). If you do this, you get the "formula" for matrix multiplication.

Let V be an n-dimensional vector space. Let {e_1, e_2, e_3, ..., e_n} be a basis for V.

That means any v in V can be written

    v = a_1*e_1 + a_2*e_2 + ... + a_n*e_n

Let f: V → V be a linear map. Let's see what it means to apply f to v. We have

    f(v) = f(a_1*e_1 + a_2*e_2 + ... + a_n*e_n)
         = f(a_1*e_1) + f(a_2*e_2) + ... + f(a_n*e_n)
         = a_1*f(e_1) + a_2*f(e_2) + ... + a_n*f(e_n)

In other words, if we know the values of f(e_1), f(e_2), ..., f(e_n) then we can calculate the value of f(v) for any v.

Every choice of value for f(e_1), f(e_2), ... is valid and determines a unique linear map.

For n=2, use the standard basis where e_1 is (1,0) and e_2 is (0,1).

Write f(1,0) as (a,c) and write f(0,1) as (b,d). This is just "relabeling" the values f(1,0) — under the standard basis we know that f(1,0) looks like (a,c) for some values of (a,c).

Write v = (x,y) so that

    v = (x, y)
      = x*(1,0) + y*(0,1)
      = x*e_1 + y*e_2

Then we have:

    f(v) = f(x, y)
         = x*f(e_1) + y*f(e_2)
         = x*f(1,0) + y*f(0,1)
         = x*(a,c) + y*(b,d)
         = (x*a, x*c) + (y*b, y*d)
         = (x*a + y*b, x*c + y*d)

Or written using matrix notation:

    [ a b ] [x]   [x*a + y*b]
    [ c d ] [y] = [x*c + y*d]

So that's the "formula" for applying a matrix to a single vector. It's determined entirely by the four values (a,b,c,d), but really it's determined by the action of the map on the basis vectors.

Now let f,g be linear maps and calculate g(f(v)) in the same way. You'll get the "formula" for matrix multiplication.

In other words, matrix notation, matrix multiplication formulas, etc. are "just" compact ways of representing the behavior of linear maps.

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.

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.

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.

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.

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

> 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.

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.

Where does your book assume its audience is?

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..

I like this course by Philip Klein of Brown University for a computational approach: 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.

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.

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.