The Little Book of Linear Algebra

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.

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

This is only beautiful if you already understand monoids, magmas and abelian half groups (semigroups) and how they form groups. Also, we do not talk of linear transformations, we talk of group homomorphisms.

I don't know about anyone else, but I was taught linear algebra this way in the first semester and it felt like stumbling in a dark room and then having the lights turned on in the last week as if that was going to be payback for all the toe stubbing.