Epistemic status: A brisk walkthrough of (what I take to be) the highlights of this book's contents.

The big one for mathematically understanding ML!

The idea responsible for getting me excited about linear algebra is:

Linear maps are the homomorphisms between vector spaces.

Linear algebra is about the tripartite relationship between (1) homomorphisms^[1] between vector spaces, (2) sets of equations, and (3) grids of numbers.

However, grids of numbers ('matrices'), the usual star of the show in a presentation of linear algebra, aren't foregrounded in this book. Instead, this is a book chiefly treating the homomorphisms ('linear maps') themselves, directly.

Contents and Notes

1. Vector Spaces

Vector spaces are fairly substantial mathematical structures, if you're pivoting out of thinking about set theory! Intuitively, a vector space is a space $R^{n}$ for which (1) ray addition and (2) scaling rays (emanating from the origin out to points)^[2] are both nicely defined.

Precisely, a vector space is a set $V$ defined over a field $F$ ^[3] in which

V is closed under vector addition, and vector addition is commutative, associative, there is an additive identity $\to 0$ , and there is an additive inverse for every vector $\to v \in V$ ;
V is closed under scalar multiplication, scalar multiplication is associative, and there is a multiplicative identity $1$ ;
and vector addition and scalar multiplication are connected by distribution such that, for all $a, b \in F$ and $\to v, \to x \in V$ ,^[4]

a (\to v + \to x) = a \to v + a \to x

(a + b) \to v = a \to v + b \to v

A subspace $S$ of a vector space $V$ is any subset $S \subset V$ that is still itself a vector space, under the same two operations of $V$ . Vector spaces can be decomposed into their subspaces, where you think of adding vectors drawn the different subspace via their common addition operation.

2. Finite-Dimensional Vector Spaces

You live at the origin of $R^{3}$ , and your tools are the vectors that emanate out from your home. Because we have both vector addition and scalar multiplication, we have two ways of extending (or shortening) any single vector out from the origin arbitrarily far. If we're interested in reaching points in $R^{3}$ , one immediate way to get to points we didn't have a vector directly to... is by extending a too-short vector pointed in the right direction! Furthermore, because we can always multiply a vect...

Math Upskilling Notes