I am currently preparing a talk for undergraduates during which I will have to resort to the Cauchy-Binet Formula. However, I don't expect most of the audience to be familiar with this identity and, since I would like to make my talk as self-contained as possible, I feel I should at least outline some proof. As I have very limited time, I would like that proof to be as short and quickly explainable as possible.
Does anyone know a quick way to prove this identity? Any help would be much appreciated.