I'm trying to prove the following theorem, which seems straightforward enough, but I'm confused about the wording and proving the converse:
Let T be a linear operator on a finite-dimensional vector space V, and let B be an ordered basis for V. Prove that t is an eigenvalue of T iff t is an eigenvalue of the matrix representation of $T_B$. (Sorry; I don't know how to make the subscript of B with [T].)
Here's my confusion: don't we have to have that V is a vector space consisting of column vectors in order for us to even speak of eigenvalues and eigenvectors of matrices? This isn't specified in the theorem.
Moreover, even if this were specified, I'm not sure how to get from the matrix multiplication form back to the linear operator form with my proof. Any suggestions?