Hi, I am TA'ing a graduate course on linear algebra this semester, and we have come up to a theorem that I cannot prove myself without resorting to Schur's Lemma and going into issues of density of invertible matrices, which is too advanced at this point in the class. The problem is as follows:

Let be a linear operator on the finite-dimensional vector space over ; let . Then, if for all bases of , the matrix representation of is the same, then , for some .

Of course, I should have asked sooner, but the class meets in 12 hours from now =P So if anyone can wheel off an elementary proof pretty quickly, it would be awesome.