The subtle difference between matrices and linear operators

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!

Let L be a linear transformation from vector space U to vector space V. If $\{u_1, u_2, ..., u_n}$ is a basis for U and ${v_1, v_2, ..., v_m} is a basis for V. Apply L to each [itex]u_i$ in turn and write it as a linear combination of the V basis. The coefficients give the ith column of the matrix representation of L.
But to go the other way, you don't have to say "view matrix A as a linear transformation". An n by m matrix is a linear transformation from vector space $R^n$ to $R^m$.