How can I show that if $\{v_1, v_2, \dots, v_k\}$ is a linearly dependent set, and $T$ is a linear transformation, then $\{T(v_1), T(v_2), \dots,T(v_k)\}$ is linearly dependent.
Also, Let $ T : R^n \to R^m $ be a linear transformation, and let $V$ be a subspace of $R^m$. Show that the preimage $T^{-1}(V)$ is a subspace of $R^n$.
I know that it must satisfy $T(x+y)= T(x)+T(y)$ and $T(cx)=cT(x)$. But I'm not sure where to go with this?
Best Answer
Hint: If $v_1,\dots, v_k$ is linearly dependent, that means we can find scalars $c_1,\dots,c_k$ such that $$c_1v_1+\dots+c_kv_k=0$$ and not every $c_i$ is $0$. What do you get when you apply $T$ to both sides of the equation above?