Let V be a vector space and let S = {v,…, Vn} be a set of vectors in V. Prove if the elements of S are linearly dependent, then there exists a vector in S which is a linear combination of the other elements in S.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let \( V \) be a vector space and let \( S = \{ v_1, \cdots, v_n \} \) be a set of vectors in \( V \). Prove if the elements of \( S \) are linearly dependent, then there exists a vector in \( S \) which is a linear combination of the other elements in \( S \).
Transcribed Image Text:Let \( V \) be a vector space and let \( S = \{ v_1, \cdots, v_n \} \) be a set of vectors in \( V \). Prove if the elements of \( S \) are linearly dependent, then there exists a vector in \( S \) which is a linear combination of the other elements in \( S \).
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