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5.8
Vector Space Quotients
We conclude this chapter with the construction of the quotient of a vector
space V by a subspace W . This construction requires that we first introduce
the general notion of a quotient space, which is based on the concept of an
equivalence relation. The particular vector space we are going to construct
is denoted by V /W . Its elements are called cosets. The reader will see that
in a certain sense, the vector space V /W can be thought of as the result of
subtracting W from V (not dividing V by W ), but not too much should be
read into this. The vector space V /W is defined abstractly and hence cannot
be pictured geometrically. For example, V /W isn’t given as a subspace of
some other natural vector space as most of the examples we have considered
previously are.
5.8.1
Equivalence Relations and Quotient Spaces
The first step in defining V /W is to introduce the concept of an equivalence
relation on a set. As the reader will see, this notion equivalence is simply
a generalization of the notion of equality. First, we need to recall what a
relation on a set is.
Definition 5.12. Let S be a nonempty set. A subset E of S × S is called
a relation on S. If E is a relation on S, and a and b are elements of S, we
will say a and b are related by E and write aEb if and only if (a, b) ∈ E.
A relation E on S is called an equivalence relation when the following three
conditions hold for all a, b, c ∈ S:
(i) (E is reflexive) aEa,
(ii) (E is symmetric) if aEb, then bEa, and
(iii) (E is transitive) if aEb and bEc, then aEc.
If E is an equivalence relation on S and a ∈ S, then the equivalence class of
a is defined to be the set of all elements b ∈ S such that bEa. An element
of an equivalence class is called a representative of the class.
It follows from (ii) and (iii) that any two elements in an equivalence class
are equivalent.
Example 5.30. An obvious example of an equivalence relation on an arbitrary set S is equality. That is, sEt if and only if s = t. The equivalence
classes consist of the singletons {s}, as s varies over S.
!
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Example 5.31. We will now give an equivalence relation on Z. Let us say
that two integers m and n are equivalent if and only if their difference is
even. This equivalence relation is the subset E of Z × Z consisting of the
pairs (m, n) such that m − n = 2k for some integer k. In other words, m
is equivalent to n if and only if m ≡ n mod(2). There are two equivalence
classes, the class of even integers represented by 0 and the class odd integers
represented by 1.
!
Proposition 5.31. Let E is an equivalence relation on a set S. Then every
element a ∈ S is in an equivalence class, and two equivalence classes are
either disjoint or equal. Therefore S is the disjoint union of the equivalence
classes of E.
Proof. Every element is equivalent to itself, so S is the union of its equivalence classes. We have to show that two equivalence classes are either equal
or disjoint. Suppose C1 and C2 are equivalence classes, and let a ∈ C1 ∩ C2 .
By definition, every element of C1 is equivalent to a, so C1 ⊂ C2 since
a ∈ C2 . Similarly, C2 ⊂ C1 , hence they are equal.
Definition 5.13. The set of equivalence classes in S of an equivalence relation E is called the quotient of S by E. The quotient will be denoted as
S/E.
5.8.2
Cosets
Suppose V be a vector space over F and let W be a subspace. We are now
going to use W to define an equivalence relation on V whose equivalence
classes are called em cosets of W . The set of equivalence classes of this
equivalence relation will be denoted as V /W , and the main result will be
that V /W can be made into a vector space. The definition is given in the
following Proposition.
Proposition 5.32. Given v and y in V , let us say that vEW y if and only
if v − y ∈ W . Then EW is an equivalence relation on V .
Proof. Clearly vEW v since v − v = 0 ∈ W . If vEW y, then yEW v since W
is closed under scalar multiplication, and (y − v) = (−1)(v − y). Finally,
if vEW y and yEW z, then vEW z. For v − z = (v − y) + (y − z) and W is
closed under addition. Hence EW is an equivalence relation on V .
Definition 5.14. Let v ∈ V be fixed. Then the coset of W containing v is
defined to be the set
v + W = {v + w | w ∈ W }.
(5.17)
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The notion of a coset is nothing complicated. For example, if V = R3
and W is a plane through 0, then the coset v + W is simply the plane
through v parallel to W .
Proposition 5.33. The equivalence classes of the equivalence relation EW
on V are precisely the cosets of W . In particular, v + W = y + W if and
only if v − y ∈ W .
Proof. Let C denote the equivalence class of v and consider the coset v+W .
If yEW v, then y − v = w ∈ W . Hence y = v + w, so y ∈ v + W . Therefore
C ⊂ v + W . Arguing in reverse, we also conclude that v + W ⊂ C.
As mentioned above, we will denote the quotient space (i.e. the set of
cosets of W ) by V /W . Our primary goal is to show that any two cosets can
be added and that V /W has a well defined scalar multiplication. Given two
cosets (v + W ) and (y + W ), define their sum by putting
(v + W ) + (y + W ) = (v + y) + W.
(5.18)
In order that this addition be a binary operation on V /W , we have to show
that the rule (5.18) is independent of the way we write a coset. That is,
suppose v + W = v! + W and y + W = y! + W . Then we have to show that
(v + y) + W = (v! + y! ) + W . But this is so if and only if
(v + y) − (v! + y! ) ∈ W,
which indeed holds since
(v + y) − (v! + y! ) = (v − v! ) + (y − y! ),
due to the fact that W is a subspace. Therefore, addition on V /W is well
defined. Scalar multiplication on cosets is defined by
a(v + W ) = av + W.
(5.19)
A similar argument shows that this scalar multiplication is also well defined.
We have therefore defined the quotient vector space V /W and prove one
of its basic properties.
Theorem 5.34. Let V be a vector space over a field F and suppose W is
a subspace of V . Then the set V /W of cosets of W in V with addition and
scalar multiplication defined as in (5.18) and (5.19) is a vector space over F.
If V is finite dimensional, then
dim V /W = dim V − dim W.
(5.20)
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Proof. The fact that V /W satisfies the vector space axioms is straightforward, so we will omit most of the details. The zero element is 0+W , and the
additive inverse −(v + W ) of v + W is −v + W . Properties such as associativity and commutativity of addition follow from corresponding properties
in V .
To check the dimension formula (5.20), choose a basis w1 , . . . , wk of W ,
and extend this to a basis
w1 , . . . , wk , v1 , . . . , vn−k
of V . Then I claim the cosets v1 + W, . . . , vn−k + W are a basis of V /W .
To see they are independent, put vi + !
W = αi if 1 ≤ i ≤ n − k, and suppose
there exist a1 , . . . , an−k ∈ F such that n−k
i=1 ai αi = 0 + W . This means that
!n−k
a
v
∈
W
.
Hence
there
exist
b
,
.
.
. , bk ∈ F such that
1
i=1 i i
n−k
"
i=1
ai vi =
k
"
bj wj .
j=1
But the fact that the vi and wj comprise a basis of V implies that all ai
and bj are zero. Therefore α1 , . . . , αn−k are independent. We leave the fact
that they span V /W as an exercise.
Cosets will reappear in Chapter 6 in the discussion of standard decoding
tables for linear codes.
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Exercises
Exercise 5.70. Prove that the cosets α1 , . . . , αn−k defined in the proof of
Theorem 5.34 span V /W .
Exercise 5.71. Let V be a vector space over a finite field Fq , where q = pn ,
and let W be a subspace. How many cosets does W have?
Exercise 5.72. Let W be the subspace of V = V (4, 2) spanned by 1001,
1101, and 0110. Write down all elements of W , and find a complete set of
coset representatives for V /W . That is, find an element in each coset.
Exercise 5.73. Let A and B be arbitrary subsets of a vector space V over
F. Define their Minkowski sum to be
A + B = {x + y | x ∈ A, y ∈ B}.
Show that if A and B are cosets of a subspace W of V , then so is A + B.
Exercise 5.74. Let V and W be any two subspaces of Fn .
(i) Find a formula for dim(V + W )/W .
(ii) Are the dimensions of (V + W )/W and V /(V ∩ W ) the same?
Exercise 5.75. Find a basis of the quotient R4 /W , where W is the subspace
of R4 spanned by (1, 2, 0, 1) and (0, 1, 1, 0).
Exercise 5.76. Let V be a vector space over Fp of dimension n, and let W
be a subspace of dimension k.
(1) Show that every coset of W has pk elements. (Hint: find a bijection
from W to x + W .)
(2) Show that the number of cosets of W is p(n−k) .