If A is a projection matrix then A' is also a projection matrix

Question

Is this correct? One projects to the column space of A while the other to the row space of A?

Answer 1

I presume that by $A'$ you mean $A^\top$, the transpose of $A$. Do you know that projection matrices (projecting orthogonally onto their range or column space) are characterized in general by the two criteria $$A^2=A \qquad\text{and}\qquad A=A^\top?$$ If so, you only need to check that $A^\top$ satisfies these criteria if (and only if) $A$ does. And, of course, it's obvious since $A^\top=A$.

For non-orthogonal projections, the criterion is merely $A^2=A$, and so you once again must check that one condition for $A^\top$, which is still easy.