Prove that any finite subset of a linearly independent set is linearly independent

linear-algebra

My problem: Prove that a set of vectors $S$ is linearly independent if and only if any finite subset of $S$ is linearly independent.

I tried like this:

Suppose S is LI.Then the vector $0$ cannot be expressed as a linear sum of all elements of $S$.

How it follows that a finite subset is also LI from this fact. I think $S$ can be finite or infinite.

This is a question from the book Linear Algebra - Friedberg et al.


Solution 1:

Hint: State what it means for a set of (possibly infinite) vectors to be linearly independent

Hint: Vector addition is done on a finite set. The idea of convergence of infinite sums requires an inherent topology which may not be present.

Solution 2:

A set $S$ being linearly independent by definition means no non-trivial linear combination of elements in $S$ is zero. And a linear combination is a FINITE sum.Therefore if all finite subsets are LI, $S$ is LI.

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