null space

1. Why does \(null(A) = null(A^T A)\)?

one direction: \(Ax = 0 \Rightarrow A^TAx = A^T0 = 0\)
other direction: \(A^TAx = 0 \Rightarrow xA^TAx = 0 \Rightarrow ||Ax||^2 = 0 \Rightarrow ||Ax|| = 0 \Rightarrow Ax = 0\)
see stackoverflow here