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.
Best Answer
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.