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Proving a spanning set is the rangespace

  • #1

Homework Statement


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


Homework Equations


None


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?
 

Answers and Replies

  • #2
quasar987
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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.
 
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