Let $V , W$ be two vector spaces and $f : V \rightarrow W$ a linear map. Let $w_1,...,w_n$ be elements of $W$ that are linearly independent and let $v_1,...,v_n$ be elements of V such that $f(v_i) = w_i$ for $i=1,...,n.$ Prove that $v_1,...,v_n$ are linearly independent.
This is the actual proof, but i'm not sure if its complete:
I will prove this by contradiction.
Suppose $V_a,V_b\in V $ such that they are linearly dependent vectors. then $F(V_a)=W_b$ $V_b=\alpha V_a$ then: $f(V_b)=f(\alpha V_a)=\alpha f(V_a)=\alpha W_a$
Since $W$ has independent vectors, this results as False. So by contradiction we know that $V$ has independent vectors