How to show that $\dim\left(\operatorname {Im}A\right)=\dim\left(\operatorname {Im}A^*\right)$?

Question

Let $A:\mathbb{R}^m\to\mathbb{R}^n$ be a linear map and $A^*:\mathbb{R}^n\to\mathbb{R}^m$ be the adjoint of $A$ (that's $\langle Ax,y\rangle=\langle x,A^*y\rangle$ for all $x\in\mathbb{R}^m,y\in\mathbb{R}^n$).

I know (see here) that $b\in\operatorname {Im}A$ if, and only if, $\langle b,z\rangle=0$ for all $z\in\operatorname {Ker} A^* $.

But the problem asks to conclude that $\dim\left(\operatorname {Im}A\right)=\dim\left(\operatorname {Im}A^*\right)$. How can I do this? Is it obvious?

Thanks.

Answer 1

One way to do this is to pick an orthonormal basis, and write down the matrices of $A$ and of $A^*$. One is the transpose of the other, so they have the same rank.