每週問題 December 16, 2013

這是判定線性獨立集的基本觀念問題。

Let \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n be vectors in a vector space \mathcal{V} and \mathbf{v}_1\neq\mathbf{0}. Prove that \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n are linearly dependent if and only if there exists an integer k, 1<k\le n, such that \mathbf{v}_k is a linear combination of \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_{k-1}.

 
參考解答:

如果 \{\mathbf{v}_1,\ldots,\mathbf{v}_n\} 是一線性相關集,則存在不全為零的數組 c_1,\ldots,c_n 使得

\displaystyle  c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n=\mathbf{0}

k=\max\{i\vert c_i\neq 0, 1\le i\le n\}。因為 \mathbf{v}_1\neq\mathbf{0},可知 c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k=\mathbf{0}c_k\neq 0,且 k>1。所以,

\displaystyle  \mathbf{v}_k=-\frac{1}{c_k}\left(c_1\mathbf{v}_1+\cdots+c_{k-1}\mathbf{v}_{k-1}\right)

相反方向的論證十分明顯。

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