Suppose I have an $n\times n$ real (or complex) matrix of rank $k$, and I want to pick $k$ linearly independent rows from it. I want to do this in a continuous fashion as the matrix varies continuously. I'm being a bit vague here, but I think it doesn't matter, because a colleague tells me that it's a standard fact that no matter how one tries to make this precise, the task is impossible (at least if $2\le k\le n-1$). Unfortunately he can't remember where he read this or how to prove it. Can someone confirm this and supply the missing details?
EDIT: As pointed out by several people, I shouldn't have said "rows" but rather a basis for the row space. Sorry for the confusion.