Help with linear transformation / subspace proof

Q: Let $\displaystyle T:V\rightarrow{W}$ be a linear transformation, and let $\displaystyle U$ be a subspace of $\displaystyle W$. Prove that the set $\displaystyle T^{-1}(U)=\{\vec{v}\in{V} | T(v)\in{U}\}$ is a subspace of $\displaystyle V$. What is $\displaystyle T^{-1}(U)$ if $\displaystyle U=\{\vec{0}\}$ ?

A: I dont have much, so I will show my train of thought...

First, I want to show $\displaystyle T^{-1}(U)$ is nonempty. I can do this by simply noting $\displaystyle U$ *is *a subspace; therefore, $\displaystyle T^{-1}(U)$ is nonempty. I don't know if I am reading the set correctly. Do I wan't to show the range of $\displaystyle T^{-1}(U)$ is nonempty?

I am confused by what $\displaystyle T^{-1}(U)$ represents.

Does it mean:

$\displaystyle T(\vec{v})=c_{1}T(\vec{v_{1}})+c_{2}T(\vec{v_{2}}) +

...+c_{k}T(\vec{v_{k}})\in{U}$

$\displaystyle T^{-1}(T(\vec{v}))=\vec{v}=c_{1}\vec{v_{1}}+c_{2}\vec{ v_{2}}+

...+c_{k}\vec{v_{k}}\in{V}$

???

Thanks