I think you mean a linear transformation, not a function. They're not the same thing.

I haven't thought about this question, but I have a notion. By the rank-nullity theorem, the rank would have to be zero in that case so the nullity would be n (where n is the number of columns). Any pivot column would mean a rank greater than 0. So there must be no pivot columns. The only way I can picture that is a matrix with all zeroes.