Theorem 7.3. Let $A$ be an $m\times n$ matrix. Then the equation $$A\mathbf x = \mathbf b$$ has a solution for every $\mathbf b \in \Bbb R^m$ if and only if the dual equation $$A^T\mathbf x = \mathbf 0$$ has a unique (only the trivial) solution. (Note, that in the second equation we have $A^T$, not $A$).
I have some trouble to prove this theory (In book "Linear Algebra Done Wrong" page 61). I learned about fundamental subspace of matrix and the rank theories, but I fail to link them together. Can I get some hint for this one?
Thanks.