# Proving a spanning set is the rangespace

1. May 11, 2009

### Jennifer1990

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
None

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. May 11, 2009

### quasar987

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.