I have three sets and I should proof as they are subspaces of a vectorspace.

I write down the task and than I write my ideas and problems.

Let k be a field, V a k-vectorspace and U,W subspaces of V. Are this sets subspaces of V?

$\displaystyle \alpha \cdot U = \lbrace \alpha \cdot v | v \in U \rbrace , \alpha \in k$

$\displaystyle \alpha + U = \lbrace \alpha + v | v \in U \rbrace , \alpha \in V$

$\displaystyle U \cap W, U \cup W, U + W = \lbrace v_{1} + v_{2} | v_{1} \in U, v_{2} \in W \rbrace$

Okay to the first set. I think I have to show three things for every set which are valid if we have a subvectorspace.

$\displaystyle U \neq \lbrace \rbrace$

$\displaystyle a,b \in U \Rightarrow a+b \in U$

$\displaystyle \alpha \in k, a \in U \Rightarrow \alpha a \in U$

Okay, first I have to show that the set is not empty. This means there must exist minimum one element which is not the empty set itself. Well:

$\displaystyle \alpha \cdot U \neq \lbrace \rbrace \Rightarrow \exists \alpha \in U : \alpha \neq \lbrace \rbrace$

Is this one line enough? If it is not, how to complete the argument that the first set is not empty?
Another idea was that A vectorspace has a abelian group which have a neutral element and if I found here a neutral element I have shown it but the neutral element of addition is here the empty set and I am not allowd to use it or?

thanks for help