For example, if I were to prove that all symmetric matrices are diagonalizable, may I say "view symmetric matrix A as the matrix of a linear operator T wrt an orthonormal basis. So, T is self-adjoint, which is diagonalizable by the Spectral thm. Hence, A is also so." Is it a little awkward to specify a basis in the proof? Are linear operators and matrices technically two different classes of objects that may be linked by some "matrix representation function" wrt a basis? Thanks!