## 每週問題 October 12, 2015

(a) Let $A$ be an $m\times n$ matrix and $B$ be an $m\times p$ matrix. Show that

$C(A)+C(B)=C\left(\begin{bmatrix} A&B \end{bmatrix}\right)$,

where $C(X)$ denotes the column space of $X$.

(b) Let $P$ be an $m\times n$ matrix and $Q$ be a $p\times n$ matrix. Show that

$N(P)\cap N(Q)=N\left(\begin{bmatrix} P\\ Q \end{bmatrix}\right)$,

where $N(X)$ denotes the nullspace of $X$.

(a)

\displaystyle\begin{aligned} \mathbf{x}\in C(A)+C(B)&\Leftrightarrow \mathbf{x}=A\mathbf{y}+B\mathbf{z},~\mathbf{y}\in\mathbb{C}^n,~\mathbf{z}\in\mathbb{C}^p\\ &\Leftrightarrow \mathbf{x}=\begin{bmatrix} A&B \end{bmatrix}\begin{bmatrix} \mathbf{y}\\ \mathbf{z} \end{bmatrix},~\mathbf{y}\in\mathbb{C}^n,~\mathbf{z}\in\mathbb{C}^p\\ &\Leftrightarrow \mathbf{x}\in C\left(\begin{bmatrix} A&B \end{bmatrix}\right) \end{aligned}

(b)

\displaystyle\begin{aligned} \mathbf{x}\in N(P)\cap N(Q)&\Leftrightarrow P\mathbf{x}=\mathbf{0},Q\mathbf{x}=\mathbf{0}\\ &\Leftrightarrow \begin{bmatrix} P\\ Q \end{bmatrix}\mathbf{x}=\mathbf{0}\\ &\Leftrightarrow \mathbf{x}\in N\left(\begin{bmatrix} P\\ Q \end{bmatrix}\right) \end{aligned}