Let $V$ and $W$ be vector spaces over a field $F$ and let $T : V \rightarrow W$ be a linear map. Let $W_0$ be a subspace of $W$ and let $V_0 = \{v \in V | T(v) \in W_0\}$ . Prove that $V_0$ is a subspace of $V$.
I'm not sure how to tackle this problem
On
For all subspaces $\{0\}$ is contained in every subspace therefore $0\in W_0$ on the other hand $\{0\}\in Ker(T)$ and therefore $\{0\}\in V_0$ as $T(0_{V_0})=0_{W_0}$
Let there be $v_1,v_2\in V_{0}$ and $\alpha \in F$
Using terms $2,3$ (as Dace mentioned) at once we get: $$T(\alpha v_1+v_2)=T(\alpha v_1)+T(v_2)=\alpha T(v_1)+T(v_2)=\alpha w_1+w_2\in W_0$$
As $W_0$ is a subspace and closed to linear combinations
Certainly $V_0$ is a subset of $V$. To show $V_0$ is a subspace of $V$, we just have to show the following: $$\begin{align}&(1)\qquad 0\in V_0\\&(2)\qquad\text{for all $u,v\in V_0$ we have $u+v\in V_0$}\\&(3)\qquad\text{for all $c\in F$ and all $v\in V_0$ we have $cv\in V_0$}\end{align}$$ You will need to use the fact that $T$ is a linear operator to show these.