## 每週問題 December 30, 2013

Suppose the vector spaces $\mathcal{V}$ and $\mathcal{W}$ have bases $\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}$ and $\{\mathbf{w}_1,\ldots,\mathbf{w}_n\}$, respectively. Show that there is exactly one linear transformation $T:\mathcal{V}\to\mathcal{W}$ with the property $T(\mathbf{v}_i)=\mathbf{w}_i$, for $i=1,\ldots,n$.

$\displaystyle T(\mathbf{x})=c_1\mathbf{w}_1+\cdots+c_n\mathbf{w}_n$

$\mathbf{x}=\mathbf{v}_i$ 代入上式，可得 $T(\mathbf{v}_i)=\mathbf{w}_i$$i=1,\ldots,n$。設 $\mathbf{y}=d_1\mathbf{v}_1+\cdots+d_n\mathbf{v}_n$，對於任意 $\alpha$$\beta$

\displaystyle\begin{aligned} T(\alpha\mathbf{x}+\beta\mathbf{y})&=T(\alpha c_1\mathbf{v}_1+\cdots+\alpha c_n\mathbf{v}_n+\beta d_1\mathbf{v}_1+\cdots+\beta d_n\mathbf{v}_n)\\ &=T((\alpha c_1+\beta d_1)\mathbf{v}_1+\cdots+(\alpha c_n+\beta d_n)\mathbf{v}_n)\\ &=(\alpha c_1+\beta d_1)\mathbf{w}_1+\cdots+(\alpha c_n+\beta d_n)\mathbf{w}_n\\ &=\alpha(c_1\mathbf{w}_1+\cdots+c_n\mathbf{w}_n)+\beta(d_1\mathbf{w}_1+\cdots+d_n\mathbf{w}_n)\\ &=\alpha T(\mathbf{x})+\beta T(\mathbf{y}), \end{aligned}

$T$ 為一線性變換。假設 $T'$ 為一線性變換滿足 $T'(\mathbf{v}_i)=\mathbf{w}_i$$i=1,\ldots,n$。根據 $T$$T'$ 的定義，每一 $\mathbf{x}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n$ 都有

\displaystyle\begin{aligned} T'(\mathbf{x})&=T'(c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n)\\ &=c_1T'(\mathbf{v}_1)+\cdots+c_nT'(\mathbf{v}_n)\\ &=c_1\mathbf{w}_1+\cdots+c_n\mathbf{w}_n=T(\mathbf{x}),\end{aligned}

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