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

Step 1 ：Check if the zero vector is in the set: \(\left[\begin{array}{c} 5x \\ 0 \\ 2y \end{array}\right]\) with \(x=0\) and \(y=0\) gives \(\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]\)

Step 2 ：Check if the set is closed under vector addition and scalar multiplication: \(\left[\begin{array}{c} 5x_1 \\ 0 \\ 2y_1 \end{array}\right] + \left[\begin{array}{c} 5x_2 \\ 0 \\ 2y_2 \end{array}\right] = \left[\begin{array}{c} 5(x_1+x_2) \\ 0 \\ 2(y_1+y_2) \end{array}\right]\) and \(c\left[\begin{array}{c} 5x \\ 0 \\ 2y \end{array}\right] = \left[\begin{array}{c} 5(cx) \\ 0 \\ 2(cy) \end{array}\right]\)

Step 3 ：\boxed{\text{Final Answer: (B)}}