Thread: 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
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?

Originally Posted by lllll

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.

Let and suppose this set is linearly independent. Then . Suppose there exists (in the field) such that thus . But since is one-to-one its kernel is the zero-vector thus . Thus, are linearly independent.

Would it be a similar prove if S was a basis instead of a subset?