Help with full rank factorization

learningstill

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

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

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

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.