Let $v_1, …, v_n$ be vectors in a vector space and let $T: V \rightarrow W$ be a linear transformation. Prove $\{T(v_1), … T(v_n)\}$ linearly independent $\Rightarrow \{v_1, … v_n\}$ linearly independent.
First of all, $\{T(v_1), … T(v_n)\}$ linearly independent means:
$c_1T(v_1) + … + c_nT(v_n) = 0 \Longrightarrow c_1 = … = c_n = 0$
Note $c_1T(v_1) + … + c_nT(v_n) = T(c_1v_1 + … + c_nv_n)$
I'm not sure where to go from here.
Best Answer
You can prove it by contradiction. Note that $0$ does not belong to any set linearly independent, so you can rule out any of the vectors being null. If $c_1v_1 + ... + c_nv_n = 0$, where some scalar is non-null, say $c_1 \neq 0$, then we can write $$ v_1 = \frac{-c_2}{c_1}v_2- \cdots\frac{-c_n}{c_1}v_n .$$
Applying $T$ $$ Tv_1 = \frac{-c_2}{c_1}Tv_2- \cdots\frac{-c_n}{c_1}Tv_n ,$$
Which Means that the set of images is linearly dependent.