If the Set W Is a Vector Space, Find a Set

Multiple Choice

If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space.
-Let $\mathrm { H }$ be the set of all points in the $\mathrm { xy }$ -plane having at least one nonzero coordinate: $\mathrm { H } = \left\{ \left[ \begin{array} { l } \mathrm { x } \\ \mathrm { y } \end{array} \right] : \mathrm { x } , \mathrm { y } \right.$ not both zero $\}$ . Determine whether $\mathrm { H }$ is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy:
A: Contains zero vector
B: Closed under vector addition
C: Closed under multiplication by scalars

A) $\mathrm { H }$ is not a vector space; not closed under vector addition
B) $\mathrm { H }$ is not a vector space; does not contain zero vector
C) $\mathrm { H }$ is not a vector space; fails to satisfy all three properties
D) $\mathrm { H }$ is not a vector space; does not contain zero vector and not closed under multiplication by scalars