The subtle difference between matrices and linear operators

[1230wc]

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!

 

[HallsofIvy]

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 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.<br /> (The bases do not have to be orthonormal. You just have to have some inner product defined on the space to talk about self-adjoint.)<br /> <br /> But to go the other way, you don&#039;t have to say &quot;view matrix A as a linear transformation&quot;. An n by m matrix <b>is</b> a linear transformation from vector space R^n to R^m.