# Help with full rank factorization

1. Jul 23, 2008

### learningstill

I've been tasked with proving the existence of a full rank factorization for an arbitrary m x n matrix, namely:

Let $$\textit{A}$$ $$\in$$ $$\textbf{R}^{m x n}$$ with $$\textit{rank(A) = r}$$ then there exist matrices $$\textit{B}$$ $$\in$$ $$\textbf{R}^{m x r}$$ and $$\textit{C}$$ $$\in$$ $$\textbf{R}^{r x n}$$ such that $$\textit{A = BC}$$. Furthermore $$\textit{rank(A) = rank(B) = r}$$.

I think I can prove the second property if I assume the first using $$\it{rank(AB)}$$ $$\leq$$ $$\it{rank(A)}$$ and $$\it{rank(AB)}$$ $$\leq$$ $$\it{rank(B)}$$.

I'd appreciate a push in the right direction. Thanks.

EDIT: I just realized I posted this in the wrong forum. Could a mod move this? My apologies.