Let $A∈\mathbb C^{p×n}$, and $C∈\mathbb C^{p×q}$. Show that you can not have more than one matrix $B∈\mathbb C^{n×q}$. Show that the columns of $D=B_1 – B_2$ are in $\operatorname{null}(A)$ given that $AB=C$ and $B^Hv=\mathbf0$ for every vector $v∈\operatorname{null}(A)$. Assume that $B_1$ and $B_2$ are individual matrices.
I am stumped by the 2 matrices $B_1$ and $B_2$. If they were given as bases I think I could pull something together.
I know that $n=\dim(A)+\operatorname{rk}(A)$
But I don't know how to get from $B$ to $D$
I think what you're asking is how to prove that there exists at most 1 matrix B such that AB=C. This statement is false. Consider A= $\left[\begin{array} &2&2\\0& 0\end{array}\right]$. If B$_1$= $\left[\begin{array} &0&0\\0& 0\end{array}\right]$ and B$_2$ = $\left[\begin{array} &-1&1\\1& -1\end{array}\right]$, then AB$_1$ = AB$_2$ = B$_1$.