Suppose T∈L(V,W) and v1,…,vm is a list of vectors in V such that Tv1,…,Tvm is a linearly independent list in W. Prove that v1,…,vm is linearly independent.
We are given that
a1Tv1+⋯+amTvm=0⟺every aj=0
By linearity this is the same as saying
T(a1v1+⋯+amvm)=0⟺every aj=0
Which implies v1,…,vm is linearly independent, as if it wasn't we could write 0=a1v1+⋯+amvm with some aj=0 and have T(0)=0.