Understanding Quotient Spaces in Mathematics

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QUOTIENT
SPACES
 
CHAPTER 3
 
COSETS
 
In 
mathematics
, specifically 
group theory
,
subgroup
 
H
 of a 
group
 
G
 may be used to
decompose the underlying set
of 
G
 into 
disjoint
 equal-size subsets
called 
cosets
. There are 
left
cosets
 and 
right cosets
. Cosets (both left
and right) have the same number of
elements (
cardinality
) as does 
H
.
Furthermore, 
H
 itself is both a left coset
and a right coset. The number of left
cosets of 
H
 in 
G
 is equal to the number of
right cosets of 
H
 in 
G
. This common value
is called the 
index
 of 
H
 in 
G
 and is usually
denoted by [
G
 : 
H
].
 
 
Quotient
space
 
The quotient space  of a 
topological
space
  and an 
equivalence
relation
  on  is the set
of 
equivalence classes
 of points
in  (under the 
equivalence relation
 )
together with the following
topology given to subsets of : a
subset  of  is called open 
iff
  is open
in . Quotient spaces are also called
factor spaces.
 
Suppose if A is group, and B is subgroup of A, and is an element of A, then
aB = {ab : b an element of B } is left coset of B in A,
The left coset of B in A is subset of A of form aB for some a(element of A).
In aB(left coset), a is representative of coset.
And
Ba = {ba : b an element of B } is right coset of B in A.
The right coset of 
B
 in 
A
 is subset of 
A
 of form 
Ba
 for some 
a
(element of 
A
). In
right coset 
Ba
, element 
a
 is referred to as representative of coset.
 
 
DIMENSIO
NS OF A
QUOTIENT
SPACE
 
In 
linear algebra
, the 
quotient
 of
vector space
 
V
 by a 
subspace
 
N
 is a
vector space obtained by
"collapsing" 
N
 to zero. The space
obtained is called a 
quotient space
 and
is denoted 
V
/
N
 (read "
V
 mod 
N
" or
"
V
 by 
N
").
 
Formally, the construction is as follows.
[1]
 Let 
V
 be
vector space
 over a 
field
 
K
, and let 
N
 be
subspace
 of 
V
. We define an 
equivalence relation
 ~
on 
V
 by stating that 
x
 ~ 
y
 if 
x
 − 
y
 ∈ 
N
. That is, 
x
 is
related to 
y
 if one can be obtained from the other by
adding an element of 
N
. From this definition, one can
deduce that any element of 
N
 is related to the zero
vector; more precisely, all the vectors in 
N
 get mapped
into the 
equivalence class
 of the zero vector.
 
The equivalence class – or, in this case, the 
coset
 – of 
x
 is often denoted
[
x
] = 
x
 + 
N
since it is given by
[
x
] = {
x
 + 
n
 : 
n
 ∈ 
N
}.
The quotient space 
V
/
N
 is then defined as 
V
/~, the set of all equivalence classes
over 
V
 by ~. Scalar multiplication and addition are defined on the equivalence
classes by
[2]
[3]
α[
x
] = [α
x
] for all α ∈ 
K
, and
[
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
 into a vector space over 
K
 with 
N
 being the zero class, [0].
The mapping that associates to 
v
 ∈ 
V
 the equivalence class [
v
] is known as
the 
quotient map
.
Alternatively phrased, the quotient space V/N is the set of all 
affine
subsets
 of V which are 
parallel
 to N.
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In group theory, a subgroup H of a group G helps decompose G into equal-size disjoint subsets called cosets. Quotient spaces in mathematics involve equivalence classes under a given relation and a specific topology. Furthermore, in linear algebra, the quotient of a vector space by a subspace results in a new vector space. Explore the concept of quotient spaces and their applications in various mathematical contexts.


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  1. QUOTIENT SPACES CHAPTER 3

  2. In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H]. COSETS

  3. The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in . Quotient spaces are also called factor spaces. Quotient space

  4. Suppose if A is group, and B is subgroup of A, and is an element of A, then aB = {ab : b an element of B } is left coset of B in A, The left coset of B in A is subset of A of form aB for some a(element of A). In aB(left coset), a is representative of coset. And Ba = {ba : b an element of B } is right coset of B in A. The right coset of B in A is subset of A of form Ba for some a(element of A). In right coset Ba, element a is referred to as representative of coset.

  5. In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read "V mod N" or "V by N"). DIMENSION S OF A QUOTIENT SPACE

  6. Formally, the construction is as follows.[1]Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x y N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector; more precisely, all the vectors in N get mapped into the equivalence class of the zero vector.

  7. The equivalence class or, in this case, the coset of x is often denoted [x] = x + N since it is given by [x] = {x + n : n N}. The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplication and addition are defined on the equivalence classes by[2][3] [x] = [ x] for all K, and [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 into a vector space over K with N being the zero class, [0]. The mapping that associates to v V the equivalence class [v] is known as the quotient map. Alternatively phrased, the quotient space V/N is the set of all affine subsets of V which are parallel to N.

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