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

Answer

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Answer

\boxed{\text{(A) None of them}} is the final answer since all three conditions are satisfied.

Steps

Step 1 ：Check if the zero vector is in the set: $[\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ is in the set, satisfying condition 1.

Step 2 ：Check if the set is closed under vector addition: $[\begin{matrix} 5 (x_{1} + x_{2}) \\ 0 \\ 2 (y_{1} + y_{2}) \end{matrix}]$ is in the set, satisfying condition 2.

Step 3 ：Check if the set is closed under scalar multiplication: $[\begin{matrix} 5 (c x) \\ 0 \\ 2 (c y) \end{matrix}]$ 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.

Which one of the following sets is not a vector subspace of R3?(A)None of them(B)S={[5x02y]:x,y∈R}

Answer

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