Let $T: V \to W$ be a linear transformation. Let $S = \{v_1,…,v_k\}$ and assume $T(S)$ is linearly independent. Show S is also linearly independent.
I think I just have to prove that if $a_1 v_1 + … + a_k v_k = 0$ then all $a_i$ must be 0 but am not sure how and how to use the fact that T(S) is linearly independent.
Would this be a valid proof?
$a_1 v_1 + … + a_k v_k = 0$
$T(a_1v_1 + … a_k v_k) = 0$
but T(S) is linearly independent so all $a_i$ must be 0
so $a_1 v_1 + … + a_k v_k$ is linearly independent
Thank you.
Best Answer
Your proof as it is written now is not valid, since $T(a_1 v_1 + \cdots + a_k v_k)=0$ doesn't imply that all $a_i=0$. However, since $T$ is linear you have $$ T(a_1 v_1 + \cdots + a_k v_k) = a_1 T(v_1) + \cdots + a_k T(v_k) $$ where you now have a linear combination of the elements of $T(S)$.