Vector Spaces

What and why

A vector space $V$ is a set of elements that is closed under vector addition and scalar multiplication. Vector spaces are useful because they tell us what type of elements we are dealing with, e.g. how many entries are in each vector, and which operations are defined for these vectors, e.g. the cross product is only defined for $\mathbb{R}^3$.

Subspaces

What and why

Sometimes, we only want to focus on a specific "part" of a vector space. In this case, subspaces allow us to focus only on a subset of an entire vector space, e.g. the column space or null space of a linear transformation.