Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Well, you haven't even attempted one half of the problem. – Chris Eagle Jan 16 '13 at 15:04
@ChrisEagle I have started with the only if part. – Vinod Jan 16 '13 at 15:06
Is it necessary to use all elements of $S$ when creating the linear sum that is not supposed to be $0$? If $S=\{x,y\}$, is a linear sum $ax+by$ or must is be $x+y$ only? If the former, can I take $a=b=0$? How about $a=0, b\neq 0$ or vice versa? – Dilip Sarwate Jan 16 '13 at 15:08
@DilipSarwate linear sum $ax+by$ is possible. – Vinod Jan 16 '13 at 15:10

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.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.