Rank factorization ($A=BC,B\in {\mathbb{R}}^{m\times r},C\in {\mathbb{R}}^{r\times n}$)

For a matrix of matrix rank $r$, choose an ordered basis of its column space as the columns of $B$. Express every column of $A$ in that basis and put its coordinates in $C$. Then $A=BC$, $B$ has full column rank and $C$ has full row rank. Choosing independent original columns makes the corresponding columns of $C$ the standard basis vectors, directly proving its rank.

The basis choice is not unique: for any invertible $r\times r$ matrix $S$, the factors $BS$ and $S^{-1}C$ have the same product. Once $B$ is fixed, its coordinate matrix is unique. For rank zero, the empty factors of the displayed shapes represent the zero matrix.