- Source: Quotient space (linear algebra)
In linear algebra, the quotient of a vector space
V
{\displaystyle V}
by a subspace
N
{\displaystyle N}
is a vector space obtained by "collapsing"
N
{\displaystyle N}
to zero. The space obtained is called a quotient space and is denoted
V
/
N
{\displaystyle V/N}
(read "
V
{\displaystyle V}
mod
N
{\displaystyle N}
" or "
V
{\displaystyle V}
by
N
{\displaystyle N}
").
Definition
Formally, the construction is as follows. Let
V
{\displaystyle V}
be a vector space over a field
K
{\displaystyle \mathbb {K} }
, and let
N
{\displaystyle N}
be a subspace of
V
{\displaystyle V}
. We define an equivalence relation
∼
{\displaystyle \sim }
on
V
{\displaystyle V}
by stating that
x
∼
y
{\displaystyle x\sim y}
if
x
−
y
∈
N
{\displaystyle x-y\in N}
. That is,
x
{\displaystyle x}
is related to
y
{\displaystyle y}
if and only if one can be obtained from the other by adding an element of
N
{\displaystyle N}
. This definition implies that any element of
N
{\displaystyle N}
is related to the zero vector; more precisely, all the vectors in
N
{\displaystyle N}
get mapped into the equivalence class of the zero vector.
The equivalence class – or, in this case, the coset – of
x
{\displaystyle x}
is defined as
[
x
]
:=
{
x
+
n
:
n
∈
N
}
{\displaystyle [x]:=\{x+n:n\in N\}}
and is often denoted using the shorthand
[
x
]
=
x
+
N
{\displaystyle [x]=x+N}
.
The quotient space
V
/
N
{\displaystyle V/N}
is then defined as
V
/
∼
{\displaystyle V/_{\sim }}
, the set of all equivalence classes induced by
∼
{\displaystyle \sim }
on
V
{\displaystyle V}
. Scalar multiplication and addition are defined on the equivalence classes by
α
[
x
]
=
[
α
x
]
{\displaystyle \alpha [x]=[\alpha x]}
for all
α
∈
K
{\displaystyle \alpha \in \mathbb {K} }
, and
[
x
]
+
[
y
]
=
[
x
+
y
]
{\displaystyle [x]+[y]=[x+y]}
.
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives). These operations turn the quotient space
V
/
N
{\displaystyle V/N}
into a vector space over
K
{\displaystyle \mathbb {K} }
with
N
{\displaystyle N}
being the zero class,
[
0
]
{\displaystyle [0]}
.
The mapping that associates to
v
∈
V
{\displaystyle v\in V}
the equivalence class
[
v
]
{\displaystyle [v]}
is known as the quotient map.
Alternatively phrased, the quotient space
V
/
N
{\displaystyle V/N}
is the set of all affine subsets of
V
{\displaystyle V}
which are parallel to
N
{\displaystyle N}
.
Examples
= Lines in Cartesian Plane
=Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to Y. This gives a way to visualize quotient spaces geometrically. (By re-parameterising these lines, the quotient space can more conventionally be represented as the space of all points along a line through the origin that is not parallel to Y. Similarly, the quotient space for R3 by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a plane which only intersects the line at the origin.)
= Subspaces of Cartesian Space
=Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1, ..., xn). The subspace, identified with Rm, consists of all n-tuples such that the last n − m entries are zero: (x1, ..., xm, 0, 0, ..., 0). Two vectors of Rn are in the same equivalence class modulo the subspace if and only if they are identical in the last n − m coordinates. The quotient space Rn/Rm is isomorphic to Rn−m in an obvious manner.
= Polynomial Vector Space
=Let
P
3
(
R
)
{\displaystyle {\mathcal {P}}_{3}(\mathbb {R} )}
be the vector space of all cubic polynomials over the real numbers. Then
P
3
(
R
)
/
⟨
x
2
⟩
{\displaystyle {\mathcal {P}}_{3}(\mathbb {R} )/\langle x^{2}\rangle }
is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only. For example, one element of the quotient space is
{
x
3
+
a
x
2
−
2
x
+
3
:
a
∈
R
}
{\displaystyle \{x^{3}+ax^{2}-2x+3:a\in \mathbb {R} \}}
, while another element of the quotient space is
{
a
x
2
+
2.7
x
:
a
∈
R
}
{\displaystyle \{ax^{2}+2.7x:a\in \mathbb {R} \}}
.
= General Subspaces
=More generally, if V is an (internal) direct sum of subspaces U and W,
V
=
U
⊕
W
{\displaystyle V=U\oplus W}
then the quotient space V/U is naturally isomorphic to W.
= Lebesgue Integrals
=An important example of a functional quotient space is an Lp space.
Properties
There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence
0
→
U
→
V
→
V
/
U
→
0.
{\displaystyle 0\to U\to V\to V/U\to 0.\,}
If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U:
c
o
d
i
m
(
U
)
=
dim
(
V
/
U
)
=
dim
(
V
)
−
dim
(
U
)
.
{\displaystyle \mathrm {codim} (U)=\dim(V/U)=\dim(V)-\dim(U).}
Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x in V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).
Quotient of a Banach space by a subspace
If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M by
‖
[
x
]
‖
X
/
M
=
inf
m
∈
M
‖
x
−
m
‖
X
=
inf
m
∈
M
‖
x
+
m
‖
X
=
inf
y
∈
[
x
]
‖
y
‖
X
.
{\displaystyle \|[x]\|_{X/M}=\inf _{m\in M}\|x-m\|_{X}=\inf _{m\in M}\|x+m\|_{X}=\inf _{y\in [x]}\|y\|_{X}.}
= Examples
=Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R.
If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
= Generalization to locally convex spaces
=The quotient of a locally convex space by a closed subspace is again locally convex. Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α ∈ A} where A is an index set. Let M be a closed subspace, and define seminorms qα on X/M by
q
α
(
[
x
]
)
=
inf
v
∈
[
x
]
p
α
(
v
)
.
{\displaystyle q_{\alpha }([x])=\inf _{v\in [x]}p_{\alpha }(v).}
Then X/M is a locally convex space, and the topology on it is the quotient topology.
If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M.
See also
Quotient group
Quotient module
Quotient set
Quotient space (topology)
References
Sources
Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-3-319-11079-0.
Dieudonné, Jean (1976), Treatise on Analysis, vol. 2, Academic Press, ISBN 978-0122155024
Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-90093-4.
Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
Roman, Steven (2005). Advanced Linear Algebra. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-24766-1.
Kata Kunci Pencarian:
- Subgrup normal
- Kecerdasan kolektif
- Quotient space (linear algebra)
- Quotient space (topology)
- Quotient space
- Kernel (algebra)
- *-algebra
- Quotient ring
- Linear subspace
- Symmetric algebra
- Dual space
- Normed vector space
