Interesting that this got posted yesterday, because I was thinking about it last night as well. I am far on the visual side of the visual / symbolic dichotomy. (In Peircean categories, what we really are looking at is people who think more with iconic signs versus people who think more with symbolic ones.)

I happened to think about this because I was reviewing matrix algebra, and although I could perform matrix multiplication fine simply by following the algorithm, I didn't feel I really had

Having gotten that image, that's that: until I go senile (OK, Murphy, OK, until I go

I happened to think about this because I was reviewing matrix algebra, and although I could perform matrix multiplication fine simply by following the algorithm, I didn't feel I really had

*gotten*it until until I saw the rows of the first matrix floating over to the second, rotating ninety degrees, and joining with the columns of its "mate." (Yeah, it was kind of like matrix sex.)Having gotten that image, that's that: until I go senile (OK, Murphy, OK, until I go

*completely*senile), I will never forget how that operation works again.
Matrix algebra is a weird combination of equation and visualization (for exactly the "matrix sex" reasons you talk about). I've always had a lot more trouble thinking through those problems. The nice thing is that condensed linear algebra notation often (not always) works like a "typical" equation. So you can see what the inverse is going and optimizing it by looking at it as if there weren't a bunch of vectors underneath the condensed notation (which is what I often do to think through it faster). I'm always worried, though, that that will give me the wrong intuition in some cases.

ReplyDeleteIf A is an m times n matrix and x is a column vector of length n, there are two ways of thinking about Ax. One is the way you saw the problem, which is visualizing how A acts on the x (or on multiple columns). More insightful maybe is to interpret Ax as the vector x acting on A. So if the columns of A are written a_i, and the entries of x are written x_i, we have

ReplyDeleteAx=x_1*a_1+...+x_ia_i+...+x_na_n

In other words Ax is a linear combination of the columns of A with coefficients given by the components of x.

John, that might be more insightful if you are NOT a picture kind of guy. For me, I'd have to turn it into an image before I knew what the heck you were talking about.

DeleteThere is a very clear picture of what's going on, but maybe I didn't do a good job of conveying it. If you like, take the column x, rotate it 90 degrees counter-clockwise, then multiply each x entry against the entire column of a that it is next to. Then sum of these vectors.

DeleteIt's the most natural way to look at what's going on because it gives insight into the "image" of A; i.e. what you can actually produce by multiplying A against different vectors x.

That is, you actually have a visual understanding of what A actually does. Of course the best way to really grok it is by doing a few calculations yourself.

OK, John, but now that I get your picture... that's *exactly* what I was describing!

DeleteNo, Gene, it's different. I was describing the *column-entries* of the *second* matrix (which for simplicity you should assume for now to be a column vector) rotating and multiplying the *columns* of your first matrix.

ReplyDeleteThe difference might seem at first glance to be trivial, but it is not. Of course you are still multiplying two matrices, but the formulation I am giving you gives you insight into all possible outputs of A by considering Ax as an action of x by A.

This is really quite a visual idea.

Sorry, John, I'm not getting this.

Delete3 and 4 here: http://www.math.caltech.edu/classes/ma1b-pr/09Ma1bPracMatrixmult2.pdf

ReplyDeleteThanks, John. I appreciate your efforts here. I think the basic problem is twofold:

Delete1) I only get my visual understanding from working with the material: verbal descriptions of what I ought to be seeing are almost useless to me!

2) I haven't had time to really work with what you are presenting. Until I do, I won't get it! But I hope to have that time soon.