Given X E R^(n x k) and X has linearly independent columns, and A = X(X'X)^-1 X', I want to prove rank(A) = k. (A is the projection matrix basically).
The Attempt at a Solution
I was told to show im(A) = L(X).
If i do that it would make sense since the Linear span of X is the basis of X with l.i columns, and dimL(X) = rank(X).
How do i show images though. This has to be a product of a linear transformation in order for it to be an image doesn't it? I'm really unsure at that point.