## 每週問題 January 9, 2017

Let $\mathcal{V}$ and $\mathcal{W}$ be two vector spaces over the same field. Suppose $F$ and $G$ are two linear transformations $\mathcal{V}\to \mathcal{W}$ such that for every $\mathbf{x}\in\mathcal{V}$, $G(\mathbf{x})$ is s scalar multiple (depending on $\mathbf{x}$) of $F(\mathbf{x})$. Prove that $G$ is a scalar multiple of $F$.

$F=0$，則 $G=0$。假設 $F\neq 0$。令 $\mathbf{x}\in\mathcal{V}$ 使得 $F(\mathbf{x})\neq\mathbf{0}$$G(\mathbf{x})=aF(\mathbf{x})$。對於任一 $\mathbf{y}\in\mathcal{V}$，若 $F(\mathbf{y})=\mathbf{0}$，則 $G(\mathbf{y})=aF(\mathbf{y})$。若 $F(\mathbf{y})\neq\mathbf{0}$$G(\mathbf{y})=bF(\mathbf{y})$，我們要證明 $a=b$。對於任意 $d\neq 0$，存在 $c$ 使得 $G(d\mathbf{x}-\mathbf{y})=cF(d\mathbf{x}-\mathbf{y})$。上式等號左邊為

$G(d\mathbf{x}-\mathbf{y})=dG(\mathbf{x})-G(\mathbf{y})=adF(\mathbf{x})-bF(\mathbf{y})$

$cF(d\mathbf{x}-\mathbf{y})=cdF(\mathbf{x})-cF(\mathbf{y})$