# Math Help - Elem. proof help?

1. ## Elem. proof help?

Hi, just reviewing some linear algebra this summer. Came across this in one of my exercise books, and was looking for a proof.

Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB = I(m).

[That last part reads "AB equals I sub m."]

2. Consider the linear transormation $R_A : \mathbb{R}^m \rightarrow \mathbb{R}^n : v \mapsto vA$. By definition of the rank, $R_A$ is an injection, hence is left-invertible; hence there exists a transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $T\circ R_A=1$. Writing $T=R_B$ we have $R_B\circ R_A=1$; but $R_B\circ R_A=R_{AB}$ so $AB=I_m$.