# Help with full rank factorization

by learningstill
Tags: factorization, rank
 P: 3 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.