Linear algebra is the study of vector spaces and maps between them.

A good book for linear algebra is Linear Algebra Done Wrong, by Sergei Treil. Another good book for linear algebra is Linear Algebra Done Right, by Sheldon Axler.

One famous result from linear algebra is that every square matrix satisfies its own characteristic equation. This is known as the Cayley Hamilton theorem.

One can study Linear algebra through the lens of abstract algebra. That is, one can consider all matrix equations and interpret them as group elements. In particular, every matrix can be interpreted as an element of a group.