Linear algebra over R and C

When the base field is $k = \mathbb{R}$ or $k = \mathbb{C}$, we can do analysis in the vector spaces. Well, not really hard analysis like differentiation or integration, but stuff like comparing sizes of numbers or taking the supremum. When we mix linear algebra with a bit of analysis, we get interesting consequences.

In this chapter, we will only deal with finite-dimensional vector spaces. There is a theory of infinite-dimensional topological vector spaces, and many of the things we shall discuss will be generalizable to that case. But we will not worry about such spaces. If you are interested in the infinite-dimensional case, go and study functional analysis. You will deal with spaces like the vector space of continuous functions $[0, 1] \to \mathbb{R}$.