Is the set $W=\{f(x) \in P(F): f(x)=0 \text{ or } f(x) \text{ has degree n}\}$ a subspace of $P(F)$ if $n\geq 1$?
I know that three conditions must exist in order for $W$ to be a subspace: the zero vector must exist, it must be closed under addition, and closed under scalar multiplication. I know how to do this for problems that actually give a function, but not ones more abstract. Any suggestions on how specifically to show the latter two conditions?