每週問題 October 12, 2015

這是計算兩個矩陣的行空間和與零空間交集的問題。

(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.

 
參考解答:
(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}

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