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

Answer

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Answer

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

Steps

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}}