Dominoes on a Board

Alright, time for a new puzzle. I have one I found recently I want to post, but some of the discussion first requires I post a puzzle I learned many years ago from my students in physics labs at UBC. It isn't a hard puzzle, which is why I never posted it, but the solution is very elegant, and it is one I like.

Consider an 8x8 chess board with two corner places deleted, in opposite corners. Attempt to tile this board with 2x1 dominoes. Prove or disprove whether such a tiling is possible.

Naturally such a tiling requires exactly (64-2)/2=31 dominoes to do this, as each domino will cover 2 squares and each square must be covered exactly once. Anyway, yeah, its not too tricky.