Suppose v1,…,vm is a linearly dependent list of vectors in V. Suppose also that W={0}. Prove that there exists w1,…,wm∈W such that no T∈L(V,W) satisfies Tvk=wk for each k=1,…,m.
Let w=0 be a vector in W. Our plan is to set wk=λkw in such a way that defining Tvk=wk leads to a contradiction because of dependence.
By 2.21 we can find the first dependent vector vj with
vj=a1v1+⋯+aj−1vj−1
Define w1,…,wj so that
wj=a1w1+⋯+aj−1wj−1
Then having Tvk=wk would not be a valid linear transformation. Since using it leads to a contradiction when we apply it to both sides of
vj=a1+⋯+aj−1vj−1
And use linearity.