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## Homework Statement

Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent.

## Homework Equations

## The Attempt at a Solution

I think that that it would be easier to prove the logically equivalent statement: Prove that a set S of vectors is linearly dependent if and only if there exists a finite subset of S that is linearly dependent.

First assume that a set S is linearly dependent. Then there exists a linear dependence relation among its vectors ##a_1s_1 + \cdots + a_ns_n = 0##, such that not all ##a_1,..., a_n## are zero. Since S is a subset of itself, there must exist a finite subset of S that is linearly dependent.

Second, assume that S has a finite subset S' that is linearly dependent. Since S' is a subset of S, this means that there exists a linear dependence relation among the vectors in S. Thus, S is also linearly dependent.

Does this prove the original statement? It seemed a little too easy.