Show that for every vector $c \in \Bbb{R}^n$, the vector $A^T(AA^T)^{-1}Ac$ is orthogonal to the null space of the matrix $A$.

Question

Let $A \in \Bbb{R}^{m \times n}$, $m<n,$ be a matrix of rank $m$.

Show that for every vector $c \in \Bbb{R}^n$, the vector $A^T(AA^T)^{-1}Ac$ is orthogonal to the null space of the matrix $A$.

I did this by showing that $P = A^T(AA^T)^{-1}A$ projects $c$ onto the rowspace of $A$ (which is orthogonal to $Null(A)$, but my professor said there is a simpler way that uses the definition of the dot product. I've been looking at this all week and I don't understand what he means.

Answer 1

Let $x$ be such that $Ax=0$. Then: $$(A^T(AA^T)^{-1}Ac)^Tx = c^T (AA^T)^{-1} Ax = c^T (AA^T)^{-1}0=0.$$