It'll be about linear algebra, which—as a lot of you know—is one of those subjects that's required knowledge for

just about any technical discipline, but it's also—I've noticed—generally poorly understood by

students taking it for the first time. A student might go through a class and learn how to compute

lots of things, like matrix multiplication, or the determinant, or cross products—which use the

determinant—or eigenvalues, but they might come out without really understanding why matrix

multiplication is defined the way that it is, why the cross product has anything to do with the

determinant, or what an eigenvalue really represents.

Often times, students end up well-practiced in the numerical operations of matrices, but are only

vaguely aware of the geometric intuitions underlying it all. But there's a fundamental difference

between understanding linear algebra on a numerical level and understanding it on a geometric level.

Each has its place, but—roughly speaking—the geometric understanding is what lets you judge what

tools to use to solve specific problems, feel why they work, and know how to interpret the results,

and the numeric understanding is what lets you actually carry through the application of those tools.

Now, if you learn linear algebra without getting a solid foundation in that geometric understanding,

the problems can go unnoticed for a while, until you've gone deeper into whatever field you happen to

pursue, whether that's computer science, engineering, statistics, economics, or even math itself.

Once you're in a class, or a job for that matter, that assumes fluency with linear algebra, the way

that your professors or your co-workers apply that field could seem like utter magic.

They'll very quickly know what the right tool to use is, and what the answer roughly looks like,

in a way that would seem like computational wizardry if you assumed that they're actually

crunching all the numbers in their head.

As an analogy, imagine that when you first learned about the sine function in trigonometry, you were

shown this infinite polynomial. This, by the way, is how your calculator evaluates the sine function.

For homework, you might be asked to practice computing approximations to the sine

function, by plugging various numbers into the formula and cutting it off at a reasonable point.

And, in fairness, let's say you had a vague idea that this was supposed to be related to triangles,

but exactly how had never really been clear, and was just not the focus of the course. Later on, if

you took a physics course, where sines and cosines are thrown around left and right, and people are

able to tell pretty immediately how to apply them, and roughly what the sine of a certain value is,

it would be pretty intimidating, wouldn't it? It would make it seem like the only people who are cut

out for physics are those with computers for brains, and you would feel unduly slow or dumb for

taking so long on each problem.

It's not that different with linear algebra, and luckily, just as with trigonometry, there are a

handful of intuitions—visual intuitions—underlying much of the subject. And unlike the trig example,

the connection between the computation and these visual intuitions is typically pretty

straightforward. And when you digest these, and really understand the relationship between the

geometry and the numbers, the details of the subject, as well as how it's used in practice, start to

feel a lot more reasonable.

In fairness, most professors do make an effort to convey that geometric understanding; the sine

example is a little extreme, but I do think that a lot of courses have students spending a

disproportionate amount of time on the numerical side of things, especially given that in this day

and age, we almost always get computers to handle that half, while in practice, humans worry about

the conceptual half.

So this brings me to the upcoming videos. The goal is to create a short, binge-watchable series

animating those intuitions, from the basics of vectors, up through the core topics that make up the

essence of linear algebra. I'll put out one video per day for the next five days, then after that,

put out a new chapter every one to two weeks. I think it should go without saying that you cannot

learn a full subject with a short series of videos, and that's just not the goal here, but what you

can do, especially with this subject, is lay down all the right intuitions, so that the learning you

do moving forward is as productive and fruitful as it can be. I also hope this can be a resource for

educators whom are teaching courses that assume fluency with linear algebra, giving them a place to

direct students whom need a quick brush-up.

I'll do what I can to keep things well-paced throughout, but it's hard to simultaneously account for

different people's different backgrounds and levels of comfort, so I do encourage you to readily

pause and ponder if you feel that it's necessary. Actually, I'd give that same advice when watching

any math video, even if it doesn't feel too quick, since the thinking that you do in your own time

is where all the learning really happens, don't you think?

So, with that as an introduction, I'll see you in the next video.

Captioned by Navjivan Pal Reviewed by Johann Hemmer 07/08/16