Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 5 / b

Choose $r$ independent columns of $A$ that form a basis of its column space, and put them in $B\in {\mathbb{R}}^{m\times r}$. For each column $a_j$ of $A$, let $c_j$ be its coordinate vector in that basis; put these coordinate vectors in $C\in {\mathbb{R}}^{r\times n}$. The column equations $a_j=B{c}_j$ give $A=BC$.

By construction $B$ has full column rank $r$. The columns of $C$ corresponding to the selected basis columns are the standard vectors of ${\mathbb{R}}^r$, so $C$ has full row rank $r$. This proves the requested rank factorization. When $r=0$, $A=0$ and the corresponding empty-column and empty-row factors have the required shapes and rank zero.

Not unique in general. For any invertible $r\times r$ matrix $S$,is another rank factorization, with the same full ranks. This is a change of basis in the intermediate $r$-dimensional space. For example, even $[1]=[1][1]=[2][1/2]$ has different full-rank factors. Once a particular ordered basis $B$ is fixed, its coordinate matrix $C$ is unique; it is the basis choice that creates the freedom.