Is it true that if $b\ \bot\ im(A)$ then $b\ \bot\ im(A^T)$

Question

Is it true that:

If $b\ \bot\ im(A)$ then $b\ \bot\ im(A^T)$ ?

I think vaguely remembering LA class I would say it is not true. But this seems like this is what is being implied by scribe notes for linear regression

Answer 1

The answer is no. Take for example the matrix $$ A = \pmatrix{1 & 2 \\ -1 & -2}.$$ The image of $A$ is $\Im(A)= \text{span} \pmatrix{1 \\ -1}$

Take $$b = \pmatrix{1 \\ 1}.$$ Vector $b$ belongs to $\Im(A)^{\perp}$. However, since $\Im(A^{T}) = \text{span} \pmatrix{1 \\ 2}$, $b\notin \Im(A^T)^{\perp}$.