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Vector spaces/Examples/Exercise
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Vector spaces
Check whether the following subsets of
R
2
{\displaystyle {}\mathbb {R} ^{2}}
are
linear subspaces
:
V
1
=
{
(
x
,
y
)
∈
R
2
∣
x
+
2
y
=
0
}
{\displaystyle {}V_{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x+2y=0\right\}}}
,
V
2
=
{
(
x
,
y
)
∈
R
2
∣
x
≥
y
}
{\displaystyle {}V_{2}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x\geq y\right\}}}
,
V
3
=
{
(
x
,
y
)
∈
R
2
∣
y
=
x
+
1
}
{\displaystyle {}V_{3}={\left\{(x,y)\in \mathbb {R} ^{2}\mid y=x+1\right\}}}
,
V
4
=
{
(
x
,
y
)
∈
R
2
∣
x
y
=
0
}
{\displaystyle {}V_{4}={\left\{(x,y)\in \mathbb {R} ^{2}\mid xy=0\right\}}}
.
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