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Prove that if a subset of a set of vectors is linearly dependent, then the entire...

Prove that if a subset of a set of vectors is linearly dependent, then the entire set is linearly dependent.

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Answer #1

Theorem 4.4.1. A superset of a linearly dependent set of vectors in a
vector space V over a field F is linearly dependent.
Proof. Case (i). Let S be a linearly dependent set of vectors containing
a finite number of elements a1, a2,..., an. Let T be a superset of S. Now
S being a finite subset of T and being linearly dependent, T is linearly
dependent, by definition.
Case (ii). Let S be a linearly dependent set of vectors containing an
infinite number of elements and T be a superset of S. Since S is linearly
dependent, there exists a finite subset P of S such that P is linearly
dependent. Now P being a linearly dependent finite subset of T, T is
linearly dependent, by definition.

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