35 minutes remaining $\varnothing$ 15 OF 17 QUESTIONS REMAINING Which one of the following sets is not a vector subspace of $R^{3}$ ? (A) None of them \[ S=\left\{\left[\begin{array}{c} 5 x \\ 0 \\ 2 y \end{array}\right]: x, y \in R\right\} \] \[ \lceil y-x\rceil \]

Step 1 ：Check if the zero vector is in the set: \(\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\) is in the set when x=0 and y=0.

Step 2 ：Check if the set is closed under vector addition: \(\begin{bmatrix} 5(x_1+x_2) \\ 0 \\ 2(y_1+y_2) \end{bmatrix}\) is in the set.

Step 3 ：Check if the set is closed under scalar multiplication: \(\begin{bmatrix} 5(cx) \\ 0 \\ 2(cy) \end{bmatrix}\) is in the set.

Step 4 ：\boxed{\text{Final Answer: (A) None of them}}