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Subspaces question

  1. Apr 28, 2010 #1
    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?
  2. jcsd
  3. Apr 28, 2010 #2
    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.
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