Let $A\in M_n(\mathbb R)$ and $b\in M_{n1}(\mathbb R)$ column vector. Prove $\exists x\in M_{n1}$ column vector solution to the equation $Ax=b\iff b=(\ker(A^{\tau}))^{\bot}$.
So my idea was to first assume that there is such an $x$, so I want to show that $\langle b,y\rangle=0$ for all $y\in \ker(A^{\tau})$, but I dont know how to tackle it.
This is not homework nor anything, so I dont mind if you spoil it.