To show that is a subspace of a vector space where you have two things to show,

that it is closed under scalar multiplication (for you need to show that ),

that it is closed under addition (for you need to show that ).

You need to show these two things as a subspace is a vector space contained in a bigger space, and this bigger space lends the smaller space the other properties. So all you need to prove is these two things.

So, for you need to show that and that for and .

Do you understand why this is what you need to show?

Now, does this hold?

The other two questions can be attacked similarly.