While I was reading a statistics paper, I came across one statement that I don't understand (I just have basic linear algebra knowledge).
Assume (in the context of regressions), we have a data matrix X, assuming that X is invertible and n>p. The paper states
" is an orthonormal matrix whose column space is orthogonal to that of X s.t. ": such matrix exists if . I don't understand where the last statement comes from.
I know that the nullspace of X has dimension n−rank(X)=n−p in full rank case and U is the orthonormal basis of the null space of X. But I don't get the link why U only exists, if , i.e. .