Let V be a vector space. Let S ? V such that 0 ? S. Prove that S is linearly dependent.
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Step 1: Assume that S is linearly independent. Show more…
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Prove that if $S=\left\{v_{1}, v_{2}, \ldots, v_{r}\right\}$ is a linearly dependent set of vectors in a vector space $V,$ and if $v_{r+1}, \ldots, v_{n}$ are any vectors in $V$ that are not in $S,$ then $\left\{v_{1}, v_{2}, \ldots, v_{r}, v_{r+1}, \ldots, v_{n}\right\}$ is also linearly dependent.
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