A psuedo inverse of a matrix $A$ is defined as a matrix $A^+$ that satisfies $AA^+A=A, A^+AA^+=A^+$ such that $AA^+$ and $A^+A$ are symmetric. Show that the row space of $A^+$ and the row space of $A^T$ are equal.
I have no idea how to start this. I was thinking of showing that the row space of $A^+$ is a subset of the row space of $A^T$ and vice versa. I tried like this: Let $x$ be a member of the row space of $A^+$. Then $x$ can be expressed as a linear combination of the rows of $A^+$. I used the fact that $(A^+)^T=(A^T)^+$. But I cannot proceed further. Can someone please give me some hint?