Proof that W Contains Span of Set S in Vector Space V

[newtomath]

Let S= { v1, v2, v3...vn} be a set of vectors in a vector space V and let W be a subspace of V containing S
show W contains span S.

Span is the smallest subspace (w) of vector space V that contains vectors in S

if a and b are two vectors in subspace C, then they are linear combinations of S

C is closed under addition and scalar multiples. Thus C contains all linear combos of S
Thus C contains W. So W contains S.

Can you advise if this is logical?

 

[hgfalling]

I'm a little confused by your proof. What is C? You also introduce two vectors a and b. Why two vectors?

You seem to have gone from "C contains W" to "W contains S." But W containing S is a given. They want you to show that W contains span S, which is a different thing.

Try the proof again, but be more concise. You don't need to introduce new subspaces or multiple vectors. Just consider a vector in span S, and show that it must be in W.