Which one of the following sets is not a vector subspace of $R^{3} ?$
(A)
None of them
(B)
\[
S=\left\{\left[\begin{array}{c}
5 x \\
0 \\
2 y
\end{array}\right]: x, y \in R\right\}
\]
Answer
\boxed{\text{(A) None of them}} is the final answer since all three conditions are satisfied.
Step 1 ：Check if the zero vector is in the set: \(\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\) is in the set, satisfying condition 1.
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, satisfying condition 2.
Step 3 ：Check if the set is closed under scalar multiplication: \(\begin{bmatrix} 5(cx) \\ 0 \\ 2(cy) \end{bmatrix}\) is in the set, satisfying condition 3.
Step 4 ：\boxed{\text{(A) None of them}} is the final answer since all three conditions are satisfied.
