Given an $m\times n$ matrix $\mathbf{A}$, an $n\times m$ matrix $\mathbf{C}$ is called a $\textbf{right inverse}$ of $\mathbf{A}$ if $\mathbf{AC}=\mathbf{I}_m$.
Show that if an $m\times n$ matrix $\mathbf{A}$ has a right inverse, then the column space of $\mathbf{A}$ is equal to $\mathbb{R}^m$.
Also, is the converse of the above statement true?
I am stuck when trying to find a connection between right inverse and column space. How can I prove the statement? Can someone give me some ideas?
Hint:
Consider the linear maps $u\colon\mathbf R^m\to \mathbf R^n$, and $v\colon \mathbf R^n\to \mathbf R^m$, associated to $A$ and $C$ respectively.
The condition on $A$ and $C$ means $u\circ v=1_{\mathbf R^n}$, and this implies $u$ is surjective (and $v$ injective). Now the column space is but the image of $u$.