# Thread: Relations between transformaion and vector in/dependence

1. ## 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,......,vn is in V)

Claim: If Tv1,.......,Tv2 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.

2. ## Re: Relations between transformaion and vector in/dependence

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.

3. ## Re: Relations between transformaion and vector in/dependence

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

4. ## Re: Relations between transformaion and vector in/dependence

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