Is a Set Linearly Independent if Every Finite Subset Is?

[bennyska]

Homework Statement

S is linearly independent iff every finite subset of S is linearly independent.

Homework Equations

The Attempt at a Solution

letting S be linearly independent is pretty easy. i am slightly worried about my logic for the other way though. it goes like this:

Let every finite subset of S be linearly independent. Let S not be linearly independent. Then there exists a v in S such that v is a linearly combination of finitely other vectors in S. put them into a subset. Then this finite subset is not linearly independent, contrary to our assumptions. Hence this v does not exist, and S is linearly independent.

i'm not sure if it makes sense to say v could be a linear combo of an infinite number of vectors in S (because S may be infinite).

 

[mighty2000]

i'm also not sure if i need to make a distinction between finite and infinite subsets of S. any thoughts or suggestions would be appreciated.

Hello there! Your logic seems sound to me. You are correct in saying that v cannot be a linear combination of an infinite number of vectors in S, as S is a finite set. So, your assumption that S is not linearly independent leads to a contradiction, and therefore, S must be linearly independent.

As for your question about distinguishing between finite and infinite subsets of S, I don't think it is necessary in this case. Since S is a finite set, we can simply say that every finite subset of S is linearly independent, without specifying the size of the subset.

Overall, your solution is well thought out and presented. Keep up the good work!