It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math.

But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matrices and Gaussian elimination, all the while bored out of your skull, until you have the tools to really start to approach the interesting areas.

Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks.

This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains.

It's funny because it's just a pedagogical nightmare to get students to care about any of this until one day two years later it all just makes sense.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away

In my experience it need not be like that at all.

One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these follow the definition of a linear transformation. That rotating a sum is same as summing the rotated(s).

[One may also mention that all differentiable functions (from vector to vector) are locally linear.]

Then you define adding two linear transformations using vector addition. Next you can define scaling a linear transformation. The point being that the combination can be expressed as linear transformations themself. No need to represent the vectors as R^d, geometric arrows and parallelogram rule would suffice.

Finally, one demonstrates composition of linear transformations and the fact that the result itself is a linear transformation.

The beautiful reveal is that this addition and composition of linear transformations behave almost the same as addition and multiplication of real numbers.

The addition asociates and commutes. The multiplication associates but doesn't necessarily commute. Most strikingly, the operations distributes. It's almost like algebra of real numbers !

Now, when you impose a coordinate system or choose a basis, the students can discover that matrix multiplication rule for themselves over a couple of days of playing with it -- Look, rather than maintaining this long list of linear transformations, I can store it as a single linear transformation in the chosen basis.

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just have it.

I learned from Strang, for what it's worth, which is basically LU, spaces, QR, then spectral.

I am really bad at math, for what it's worth; this is just the one advanced math subject that intuitively clicked for me.

>Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks.

Can you expand on your experience with this? I do some graphics programming so I understand that applying matrix transformations works, and I've seen the 3blue1brown 'matrices are spreadsheets' explanation (luv me sum spreadsheets), but the intuition still isn't really there. The 'incredibly deep "why matrix multiplication looks that way"' is totally lost on me.

Linear algebra was such a tedious thing to learn I skipped over it to abstract algebra and doubled back once I had some kind of minimally interesting framework to work with it against. Normally I think this is a foolish way to do things, but sometimes things are just so dull you have to take the hard route to power through at all.

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

The older I get the more convinced I am that "math is not hard; teaching math is hard".

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations.

It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix as a description of how to convert one sense of basis vectors to another set of basis vectors, and for that, you first need to understand how vectors are described in terms of basis vectors.

> This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains.

Which books or “non-standard” resources would you recommend then, that do a better job?

What I find amazing is, given how important linear algebra is to actual practical applications, high school math still goes so deep on calculus at the expense of really covering even basic vectors and matrices.

Where vectors do come up it’s usually only Cartesian vectors for mechanics, and only basic addition, scalar multiplication and component decomposition are talked about - even dot products are likely ignored.

I found the university level presentation of "Vector spaces -> Linear functions -> Matrices are isomorphic to linear functions" much more motivating than the rote mechanics I was taught in highschool, but it's hard to see if I would've had that appreciation without being taught the shitty way first.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations.

Short, simple answer to that question by Michael Penn: https://www.youtube.com/watch?v=cc1ivDlZ71U

Another interesting treatment by Math the World: https://www.youtube.com/watch?v=1_2WXH4ar5Q&t=4s

There's no impenetrable mystery here. Probably just bad teaching you experienced.

Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning.

It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".

Probability and statistics falls into that category too. It’s one of the more boring undergraduate math courses, but is mind-boggling useful in the real world.

(Basic probability / combinatorics is actually pretty cool, but both tend to be glossed over.)

I went through Khan academy on linear algebra a long time ago because I wanted to learn how to write rendering logic, for me I was implementing things as I learned them with immediate feedback for a lot of the material. Was probably the single most useful thing I learned then.

I had the opposite experience when learning linear algebra as I was also doing 3D computer graphics at the time. It was super interesting and fun. I guess you just have to find an application for it.

There was a brief time when I understood how linear algebra was used to render 3d models but that's all gone now.

I wonder if these days that would be a better starting point.

I hated precalc, but I loved proof-based linear algebra, my first university math course (linalg + multivariate calc).

The simplicity(/beauty) of matrix multiplication still irks me though, in the sense of "wow, seriously? when you work it out, it really looks that simple?"

Actually, the calculation of matrix multiplication is extremely easy to understand if you show it done with a row and column vector first. Once your students understand dot products, they will recognize matrix multiplication as the mere pairwise calculation of dot products.

I don't know, maybe it's because I read the book on the side in highschool when I was supposed to be doing something else but I really loved Linear Algebra. Once I understood what it was I used matrix operations for everything. Vector spaces are such a powerful abstraction.