## 每週問題 April 24, 2017

Let $A$ be an $m\times n$ matrix and $B$ be an $n\times p$ matrix. Prove that

$\dim (C(B)\cap N(A))=\dim C(B)-\dim C(AB)=\dim N(AB)-\dim N(B)$.

Note that $C(X)$ and $N(X)$ denote the column space and nullspace of $X$, respectively.

$\displaystyle \text{ran}\left(A_{/C(B)}\right)=\{A\mathbf{y}\vert\mathbf{y}\in C(B)\}=\{AB\mathbf{x}\vert\mathbf{x}\in\mathbb{C}^p\}=C(AB)$

\displaystyle\begin{aligned} \dim C(B)&=\dim\ker\left(A_{/C(B)}\right)+\dim\text{ran}\left(A_{/C(B)}\right)\\ &=\dim(C(B)\cap N(A))+\dim C(AB). \end{aligned}