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Problem 60: Determine whether it is a subspace of the given vector space $\V$
$\V = \R^2$, and $S$ consists of all vectors $(x,y)$ satisfying $x^2-y^2=0$

Problem 60: Determine whether it is a subspace of the given vector space $\V$
$\V = \R^2$, and $S$ consists of all vectors $(x,y)$ satisfying $x^2-y^2=0$.

Solution:
$S = \{(x, y) \in \R^2 : x^2-y^2=0\}$.
Let $(x_1,y_1), (x_2,y_2) in \S$,
Then,
$(x_1,y_1)+(x_2,y_2) = (x_1+x_2,y_1+y_2)$.
$\Rightarrow (x_1+x_2)^2 - (y_1+y_2)^2 $.
$= x_1^2+x_2^2+2x_1x_2-(y_1^2+y_2^2+2y_1y_2)$.
$=(x_1^2-y_1^2) +(x_2^2-y_2^2)+2(x_1x_2-y_1y_2) $.
$= 0+0+2(x_1x_2-y_1y_2) \ne 0$.
$(x_1,y_1)+(x_2,y_2) \notin S$
Hence, $S$ is NOT subspace of $\R^2$.

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