The quotient vector space , also called quotient space or factor space for short , is a term from linear algebra , a branch of mathematics . It is the vector space that arises as an image of a parallel projection along a sub-vector space . The elements of the quotient vector space are equivalence classes.
definition
Let it be a vector space over a field and a subspace of . By fixing
V
{\ displaystyle V}
K
{\ displaystyle K}
U
{\ displaystyle U}
V
{\ displaystyle V}
v
1
∼
v
2
:
⟺
v
1
-
v
2
∈
U
{\ displaystyle v_ {1} \ sim v_ {2} \;: \! \ iff v_ {1} -v_ {2} \ in U}
For
v
1
,
v
2
∈
V
{\ displaystyle v_ {1}, v_ {2} \ in V}
is defined on an equivalence relation .
V
{\ displaystyle V}
The vectors and are therefore equivalent if they differ by one vector . In other words: If the line through the points and to parallel are is, and equivalent.
v
1
{\ displaystyle v_ {1}}
v
2
{\ displaystyle v_ {2}}
U
{\ displaystyle U}
v
1
{\ displaystyle v_ {1}}
v
2
{\ displaystyle v_ {2}}
U
{\ displaystyle U}
v
1
{\ displaystyle v_ {1}}
v
2
{\ displaystyle v_ {2}}
The equivalence class of a point is
v
{\ displaystyle v}
[
v
]
: =
v
+
U
: =
{
v
+
u
∣
u
∈
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}
{\ displaystyle [v]: = v + U: = \ {v + u \ mid u \ in U \}}
,
clearly shows the "parallel" affine subspace . The equivalence classes are also known as secondary classes (this term comes from group theory ).
U
{\ displaystyle U}
v
{\ displaystyle v}
The quotient vector space from to is the set of all equivalence classes and is denoted by:
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{\ displaystyle V}
U
{\ displaystyle U}
V
/
U
{\ displaystyle V / U}
V
/
U
: =
{
[
v
]
∣
v
∈
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}
{\ displaystyle V / U: = \ {[v] \ mid v \ in V \}}
.
It forms a vector space if the vector space operations are defined by proxy:
[
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1
]
+
[
v
2
]
=
[
v
1
+
v
2
]
{\ displaystyle [v_ {1}] + [v_ {2}] = [v_ {1} + v_ {2}]}
λ
⋅
[
v
]
=
[
λ
v
]
{\ displaystyle \ lambda \ cdot [v] = [\ lambda v]}
for and .
v
,
v
1
,
v
2
∈
V
{\ displaystyle v, v_ {1}, v_ {2} \ in V}
λ
∈
K
{\ displaystyle \ lambda \ in K}
These operations are well-defined, i.e. independent of the choice of representatives.
properties
π
:
V
→
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/
U
,
v
↦
[
v
]
{\ displaystyle \ pi \ colon \; V \ to V / U, \; v \ mapsto [v]}
.
Is a complement of in , i.e. H. is the direct sum of and , the restriction of to is an isomorphism . But there is no canonical possibility to understand it as a subspace of .
W.
{\ displaystyle W}
U
{\ displaystyle U}
V
{\ displaystyle V}
V
{\ displaystyle V}
U
{\ displaystyle U}
W.
{\ displaystyle W}
π
{\ displaystyle \ pi}
W.
{\ displaystyle W}
V
/
U
{\ displaystyle V / U}
V
{\ displaystyle V}
If it is finite-dimensional, then the following relationship results for the dimensions:
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{\ displaystyle V}
dim
U
+
dim
V
/
U
=
dim
V
{\ displaystyle \ dim U + \ dim V / U = \ dim V}
The dual space of can be identified with those linear forms on that are identical on.
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/
U
{\ displaystyle V / U}
V
{\ displaystyle V}
U
{\ displaystyle U}
0
{\ displaystyle 0}
The homomorphism theorem states that a linear mapping has an isomorphism
f
:
V
→
W.
{\ displaystyle f \ colon \; V \ to W}
V
/
(
ker
f
)
→
i
m
f
{\ displaystyle V / (\ ker f) \ to \ mathrm {im} \, f}
between the quotient space from after the kernel from and the image from induced, d. H. the concatenation
V
{\ displaystyle V}
f
{\ displaystyle f}
f
{\ displaystyle f}
V
⟶
V
/
(
ker
f
)
⟶
i
m
f
⟶
W.
{\ displaystyle V \ longrightarrow V / (\ ker f) \ longrightarrow \ mathrm {im} \, f \ longrightarrow W}
is the same .
f
{\ displaystyle f}
Application in functional analysis
Many normed spaces created in the following way: Be a real or complex vector space and let 1 seminorm on . Then is a subspace of . The quotient space then becomes a normalized vector space with the norm .
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{\ displaystyle V}
p
{\ displaystyle p}
V
{\ displaystyle V}
U
=
{
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∈
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∣
p
(
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)
=
0
}
{\ displaystyle U = \ {v \ in V \ mid p (v) = 0 \}}
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{\ displaystyle V}
V
/
U
{\ displaystyle V / U}
[
v
]
↦
p
(
v
)
{\ displaystyle [v] \ mapsto p (v)}
More general: Let be a topological vector space that is not Hausdorffian . Then be analogous to above subspace define: . The quotient space with the quotient topology becomes a Hausdorff topological vector space.
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{\ displaystyle V}
U
=
{
v
∈
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∣
Each
0
-Environment contains
v
}
=
{
0
}
¯
{\ displaystyle U = \ {v \ in V \ mid {\ text {Each}} 0 {\ text {-Environment contains}} v \} = {\ overline {\ {0 \}}}}
V
/
U
{\ displaystyle V / U}
Examples
abstract
The spaces and thus also the Sobolew spaces are quotient vector spaces.
L.
p
{\ displaystyle L ^ {p}}
Concrete
The vector space and the one-dimensional sub-vector space are given . Then for example
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=
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2
{\ displaystyle V = \ mathbb {R} ^ {2}}
U
=
{
(
x
x
)
|
x
∈
R.
}
{\ displaystyle U = \ left \ {\ left. {\ bigl (} {\ begin {smallmatrix} x \\ x \ end {smallmatrix}} {\ bigr)} \ right | x \ in \ mathbb {R} \ right \}}
(
42
12
)
+
U
: =
{
(
42
12
)
+
u
|
u
∈
U
}
{\ displaystyle {\ bigl (} {\ begin {smallmatrix} 42 \\ 12 \ end {smallmatrix}} {\ bigr)} + U: = \ left \ {\ left. {\ bigl (} {\ begin {smallmatrix } 42 \\ 12 \ end {smallmatrix}} {\ bigr)} + u \, \ right | u \ in U \ right \}}
an equivalence class of the quotient space .
V
/
U
{\ displaystyle V / U}
Every straight line that is parallel to the bisecting line of the 1st quadrant is clearly an equivalence class:
See also
literature
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