Applications of vector spaces over the reals, i.e., $\mathbb{F}=\mathbb{R}$

Question

I am a self-studier going through Axler's "Linear Algebra Done Right" 3rd edition.

It seems that there is a substantial emphasis on complex vector spaces. There is a small chapter on "Operators on Real Vector Spaces," with a nice presentation of complexification. But on the whole, it seems that complex vector spaces are of primary importance.

I would appreciate any mention of practical applications of real vector spaces.

Thanks

Answer 1

Classical physics was using real spaces long before quantum physics famously needed complex Hilbert spaces. One famous application is phase space in Hamiltonian mechanics; this always has an even number of dimensions. Another is relativity, in which a real space is used to represent spacetime as a manifold.

Answer 2

Perhaps obvious, but the real vector space $\mathbb{R}^3$ is the world we live in, for the most part.

Coding and information theory use complex vector spaces, but also real vector spaces just as well. This has applications in data compression, for example.