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Problem 58: Determine whether it is a subspace of the given vector space $\V$
$\V = \R^4$, and $S$ is the set of all vectors of the form $(x_1,0,x_3,2)$

Problem 58: Determine whether it is a subspace of the given vector space $\V$
$\V = \R^4$, and $S$ is the set of all vectors of the form $(x_1,0,x_3,2)$.

Solution:
$S = \{x \in \R^4 : (x_1,0,x_3,2)\}$.
Let $(a,b),(c,d) \in R$.
Then $(a,0,b,2), (c,0,d,2) \in S$
Hence,
$(a,0,b,2)+(c,0,d,2) =(a+c,0+0,b+d,2+2) =(a+c,0,b+d,4) $
which implies that $(a,0,b,2)+(c,0,d,2) \notin S$
Consequently, $S$ is not closed under addition.
Hence, $S$ is NOT subspace of $\R^4$.

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