Let A be an $m \times n$ regular matrix of rank k. Consider $A$ as a module homomorphism from $\mathbb{R}^n$ into $\mathbb{R}^m$. Since $A$ is regular, there exists a matrix $G:\mathbb{R}^m \rightarrow \mathbb{R}^n$ such that $AGA = A$. Now $AG$ is an idempotent linear map on $\mathbb{R}^m \rightarrow \mathbb{R}^m$ and Range (A) = Range(AG).
By using this fact that "For any idempotent linear map $T : R^m \rightarrow R^m$, Range T is projective", Range(AG) is projective.
How to prove this "For any idempotent linear map $T : R^m \rightarrow R^m$, Range T is projective"?