# Hello, LaTeX!

acv,mathmeta

This year I've working on gaining a deeper understanding of AI and ML tools. To do this I've been studying a lot of math, specifically vector spaces (opens in a new tab). As I've been getting increasingly excited about the progress I'm making, I figured some blog posts were in my future. This seemed like a good enough reason to go on a side quest to set up equation rendering.

Fortunately, the blogging framework I'm using right now, Nextra (opens in a new tab), comes with $\LaTeX$ (nice) support built in. After migrating some configuration files, I was able get it set up and wanted to do a "little hello" world post with some equations!

So, as a proof-of-concept, here is the definition a subspace, of a core concept I've been studying in Sheldon Axler's text Linear Algebra Done Right (opens in a new tab).

### Subspaces

For a vector space $V$ over a field $F$ (nominally either $\mathbb{R}$ the real numbers or $\mathbb{C}$ the complex numbers), the subset $U \subseteq V$ is said to be a subspace of V if and only if ($\Leftrightarrow$) the following 3 conditions are met:

• Additive Identity: The additive identity $0 \in V$ is also $\in U$. That is, $0 \in U \cap V$, where $0$ is an element in $V$ such that $0 + x = x + 0 = x$ for all $x \in V$.
• Closure Under Addition: For any $x,y \in U$ we also have $x+y \in U$.
• Closure Under Scalar Multiplication: For $a \in F$ and $x \in U$ we have $ax \in U$.

Update 2023-06-25: Fixed the definition of a subspace to include the trivial case $U = V$ i.e. $U \subseteq V$ instead of $U \subset V$. Thanks Dad :)