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Problem 57: Determine whether it is a subspace of the given vector space $\V$
$\V = \R^2$, and $S$ is the set of all vectors $(x, y)$ in $\V$ satisfying $3x + 2y = 0$.

Problem 57: Determine whether it is a subspace of the given vector space $\V$
$\V = \R^2$, and $S$ is the set of all vectors $(x, y)$ in $\V$ satisfying $3x + 2y = 0$.

Solution:
$S = \{x \in \R^2 : 3x+2y=0\}$.
Let $(x_1,x_2),(y_1,y_2) \in S$.
Then $3x_1+2x_2 = 0$ and $3y_1+2y_2 = 0$
Hence,
$x + y = (x_1,x_2)+(y_1,y_2) =3x_1+2x_2 + 3y_1+2y_2 = 3(x_1+y_1) + 2(x_2+y_2)$
which implies that $(x_1+y_1,x_2+y_2) \in S$
Consequently, $S$ is closed under addition.
Let $a \in \R$ and $(x_1,x_2) \in S$, then
$ax = a(x_1,x_2) = a(3x_1+2x_2) = a0 = 0$
$=3(ax_1) + 2(ax_2) =0 $,
which implies that $(ax_1,ax_2) \in S$
Therefore $S$ is also closed under scalar multiplication.
It follows that $S$ is a subspace of $\R^2$.

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