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if I have five linearly dependent vectors in the vector space $\mathbb{R}^5$, lets call them $A_i$ for $i=1$ to $5$. can I find $C_{i,j}$ for $i=1,...,5$ and $j=1,...,5$ so that I get five new vectors $B_i: = \sum\limits_{j=1}^5 C_{i,j}A_j$ s.t. the $B_i$'s are linearly independent?

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Let $V$ the vector space spanned by $A_1,...,A_5$. By hypothesis, $\dim(V) \leq 4$. Now, necessarily $B_i \in V$ for $i=1,...,5$; therefore, the vectors $B_1,...,B_5$ are linearly dependent.

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Hah, I totally misread it. Good job! –  rschwieb Jan 22 '13 at 18:09

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