每週問題 August 8, 2016

這是生成空間的一個等價性質。

For a set of vectors S=\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}, prove that \hbox{span}(S) is the intersection of all subspaces that contain S.

 
參考解答:

\mathcal{W}=\cap_{S\subseteq\mathcal{V}}\mathcal{V},其中 \mathcal{V} 是一個子空間。我們的目標要證明 \hbox{span}(S)=\mathcal{W}。欲證明 \hbox{span}(S)\subseteq\mathcal{W},考慮 \mathbf{x}\in\hbox{span}(S),則 \mathbf{x}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n。若 \mathcal{V} 是任何一個包含向量集合 S 的子空間,則 \sum_{i=1}^nc_i\mathbf{v}_i\in\mathcal{V} (子空間具有向量加法與純量乘法封閉性),推論 \mathbf{x}\in\mathcal{W}。相反的,若 \mathbf{x}\in\mathcal{W},則 \mathbf{x}\in\hbox{span}(S),這是因為生成空間 \hbox{span}(S) 本身就是一個包含 S 的子空間。

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