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

S=(a1,a2,...,an)

V=(a1,a2,...,an,an+1,...,an+k)

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?