I want to know a general linear algebra theories that are needed in proving real symmetric matrix is diagonalizable.
I know of eigenvalues, eigenvectors, eigenspaces in the context of diagonalizability. But I haven't read about Gram–Schmidt orthogonalization and its related concepts yet.
What area of linear algebra theory should I understand in general to deal with and to prove statements like the above?
I have Friedberg's Linear Algebra text. If you have read it, you may give me the pages to read.
And if not long, give me some scratch of the proof of the statement.