The Little Book of Linear Algebra

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.

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".

But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". I stuck to it because that was what the course taught, and they wouldn't teach me multidimensional calculus without making me learn this stuff first, but it was months and years later when I suddenly found that linear functions were everywhere and I somehow magically had the tools and the knowledge to do stuff with them.