Thread: Relations between transformaion and vector in/dependence

So we have vector space V over the field F, and we have transformation T:V->V
(v₁,v₂,......,v_n is in V)

Claim: If T_v1,.......,T_v2 is linearly independent then v1,v2,......,vn is linearly independent.

The claim is true (I think) but I can only explain it in words so basically it's not a proof. Can someone give me a hint how to prove this? Do I have to use the conditions of linear transformation (that it preserve addition and multiplication with scalar) and somehow get to the conclusion that the vectors are independent too? Thanks.

Suppose for contradiction that the input vectors v_n are linearly dependent. Then let V be the matrix made up of these v_n's.

As the vectors are dependent V has a nonzero null space.
So for any x in that null space Vx = 0.
TVx = 0
so the null space of TV is at least the null space of V. It could be higher dimensional if T isn't invertible.
Thus the vectors of TV are not linearly independent.
So the vectors v_n must be linearly independent.

Hi,
Here's a direct proof that uses the definition of linear independence:

I actually found the proof before reading the posts, but thank you guys a lot