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Homework Help: Proving a spanning set is the rangespace

  1. May 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that the span {v1,...,vn} = V and let L:V-->W be an onto linear mapping. Prove that span {L(v1),...,L(v2)} = W

    2. Relevant equations

    3. The attempt at a solution
    I think for this question, we just have to show that if vi, where i is a real number, is a given vector in V, then L(vi) is a vector in W. Can someone help guide me on how to start the proof?
  2. jcsd
  3. May 11, 2009 #2


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    Since v_i is in V, of course L(v_i) is a vector in W since by definition, L take an element of V (that is to say, a vector in V) and bring it to an element of W (that is to say, a vector in W).

    According to the definition of a subset spanning a vector space, what we need to do here is to show that given any w in W, we can find real numbers a_i such that a_1L(v_1)+...+a_nL(v_n)=w.

    You have a lot of things to help you achieve this:
    1) The fact that L is linear,
    2) The fact that L is surjective,
    3) The fact that {v_i,...,v_n} spans V.
    Last edited: May 11, 2009
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