Linear transformation and indepence problem
I am having trouble with a problem and am in need of some help.
V & W are vector space, and T:V->W is linear.
T is 1 to 1 and S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent.
I have started the problem but am really uncertain of my approach. So far I have
Sum [biT(ai)] = 0 where i=1,2,...n
T[Sum bi(ai)] = 0
therefore there should exist some c such that
Sum [bi(ai)] = Sum ci(ai)
Is what I have done even correct?