Isn't this vector not non-empty because if you substitute x=0 and z=0 then the middle term will be -3.
Your sideways, over-written, fuzzy graphic is difficult to read. I think the content may be as follows:
\(\displaystyle \mbox{2. Determine whether }\, W\, =\, \left\{\, \left(\, \begin{array}{c} x\\x\, -\, 3\\z \end{array}\, \right)\, \bigg|\, x,\, z\, \in\, \mathbb{R}\, \right\}\, \mbox{ is a subspace of }\, \mathbb{R}^3.\)
However, you then ask about proving something about non-empty vectors, rather than proving something about subspaces, so I may be mistaken.
Kindly please reply with corrections and clarifications. When you reply, please explain what you are doing in attempting to answer the question in the exercise. Thank you!