Vector spaces
Crazy Vector Space
𝑪𝑽𝑺
=
{
[
x
y
]
∣
x
,
y
∈
ℝ
}
/
ℝ
Operations
[
x
1
y
1
]
⊕
[
x
2
y
2
]
=
[
x
1
+
x
2
-
2
y
1
+
y
2
]
α
⊙
[
x
y
]
=
[
α
x
-
2
α
+
2
α
y
]
Rank Nullity
R5 to R3
𝝋
(
(
x
1
x
2
x
3
x
4
x
5
)
)
=
(
2
x
1
+
x
2
-
3
x
4
3
x
1
+
x
2
-
x
3
-
4
x
4
-
x
5
-
x
1
-
x
2
-
x
3
+
2
x
4
-
x
5
)
R5 to R3
(
1
0
1
-1
0
-2
-2
1
-1
)
(
2
1
0
-3
0
3
1
-1
-4
-1
-1
-1
-1
2
-1
)
=
(
1
0
-1
-1
-1
0
1
2
-1
2
0
0
0
0
0
)
Matrix Representation
P2 to CVS
𝝋
(
p
0
+
p
1
x
+
p
2
x
2
)
=
[
p
1
+
2
2
p
2
]
With basis
b
→
1
=
1
b
→
2
=
x
b
→
3
=
x
2
and
g
→
1
=
[
3
0
]
g
→
2
=
[
2
1
]
P2 to CVS
𝝋
(
p
0
+
p
1
x
+
p
2
x
2
)
=
[
p
1
+
2
2
p
2
]
With basis
b
→
1
=
x
2
-
8
x
+
5
b
→
2
=
x
2
-
5
x
+
3
b
→
3
=
-
x
2
+
7
x
-
4
and
g
→
1
=
[
3
0
]
g
→
2
=
[
2
1
]
Change of Basis
P2
𝒊𝒅
(
p
0
+
p
1
x
+
p
2
x
2
)
=
p
0
+
p
1
x
+
p
2
x
2
With basis
b
→
1
=
1
b
→
2
=
x
b
→
3
=
x
2
and
d
→
1
=
x
2
-
8
x
+
5
d
→
2
=
x
2
-
5
x
+
3
d
→
3
=
-
x
2
+
7
x
-
4
P2
(
5
3
-4
1
0
0
-8
-5
7
0
1
0
1
1
-1
0
0
1
)
→
(
1
0
0
2
1
-1
0
1
0
1
1
3
0
0
1
3
2
1
)
CVS
𝝋
(
[
x
y
]
)
=
[
x
y
]
With basis
g
→
1
=
[
3
0
]
g
→
2
=
[
2
1
]
and
g
→
1
=
[
7
9
]
g
→
2
=
[
6
7
]
CVS
(
5
4
1
0
9
7
0
1
)
→
(
1
0
-7
4
0
1
9
-5
)