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601-quotient-spaces

# 601-quotient-spaces - k 2.5 Cosets and Quotient Spaces a“...

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Unformatted text preview: ? k 2.5 Cosets and Quotient Spaces ; a“. 3 2-5-1 Cosets ﬁt a: BL is 'a subspace of the vector space E and a: is a vector in E, we can form the sum L + {x} which we will write brieﬂy as L + 2:. Thus L+x is the set of Vectors y e E which can be written in the form y=a+x with a e L. 25 CH. 2: VECTOR SPACES Deﬁnition 1.- If L is a subspace of the vector space E, then for each a: e E, the set L+x is said to been ‘coset’ of L. The cosels deﬁned here are more usually known as linear manifolds. We will use the shorter terminology because these cosets are in fact the additive cosets of the subspace in the group theoretic sense. To avoid possible misunderstandings, we note straight away that the co- set L+x is not a subspace of E unless a: e L. Example 1. Let L be the subspace of the space G3 (Example 2.1 ;l) which ’ consists of all the vectors parallel to a ﬁxed line 9. Each coset of L consists of those vectors whose endpoints (when all vectors are drawn as arrows from a ﬁxed point 0) lie on a ﬁxed line parallel to g. All the cosets of L can be obtained in this way. L+a= Fig. 4. Example 2. IfL=E, then L is the only coset of itself. KL: {0}, then each coset consists of exactly one vector and each vector forms a coset. Every subspace L of E is a coset of itself, because obviously L=L+0. Theorem I. L+x=L if and only ifa: e L. Proof. 1. Suppose L+x=L, then 0=a+x for some a e L and :6: —a e L. 2. Suppose a: e L, then obviously L+x g L. Sii'ce a = (a —x) +3: and a—m e L for each a e L, we also have L _C_ L+x. Hence L+x=L. If :2: ¢ L, then L and L+x are disjoint. Because, if y e L n (L+a:), then 3/ e L and y=a+x where a e L and hence a: e L. Theorem 2. L+x1 =L+x2 ifcmd only ifs:1 —x2 6 L. 26 .___.._.1 __.....V . . .. %'*"“V"" “.0wa m... in 2.5: COSETS AND QUQTIENT SPACES Proof. It is easy to verify that (L+ m) + 3‘; = L + (a; + 3/) (see (1)). 1. If L+x1 =L+x2, then, adding (4x2), L+(x1—a:2)=L+0=L and hence x1 - x2 6 L. (Theorem 1). 2. Ifxl —-x2 6 L, then L+(x1—x2)=L and hence L+x1 =L+x2. If x, — x2 ¢ L, then L + (x1 — m2) is disjoint from L and therefore L + x1 and L + x2 are also disjoint. Two cosets of L are therefore either identical or disjoint. Theorem 3. Let N be a coset of a subspace L of E and lety e E. ThenN =L+y if and only éfy e N. Proof. Let N=L+x. By Theorem 2, L+x=L+y if and only if y—x e L, i.e., ifand only ify e L+x=N. Theorem 4. A subspace L is uniquely determined by a coset of L. Proof. Suppose N is a coset of two subspaces L1 and L2. Then, for every 3/ e N, N=L1+y=L2+y and, by adding —y, L1=L1+0=L2+0=L2. 2.5.2 Quotient Spaces The cosets of a subspace L of E can be added according to Deﬁnition 2.3 ;1. Thus (L+x)+(L+y) = L+(x+?/)o (1) z E (L+x)+(L+y) means that 2:01,l +x)+(a2+y) with al, a2 6 L, and therefore 2= ((11 +a2) + (at +31) 6 L+ (x + y). z e L+(x+y) means that z=a+(x+y) with a EL and therefore 2=,(0.+=v)+(a+y) 6(L+w)+(L+y). ' Similarly, it is possible to multiply cosets by scalars and then, for a750, oc(L+x) = L+ocx, (2) which can be proved in the same way as (1). (For oc=0, the equation (2) may be used as the deﬁnition of 0(L+x).) Theorem 5, I f N , N1, N 2 are cosets of a subspace L of E, then N1 +N2 and aN are also cosets of L. alternatively. Addition and multiplication by scalars are binary operations In the family JVL of all cosets of L. The question now arises as to whether l/VL is a vector space with these binary operations. In order to answer this, we must see if the axioms of Deﬁnition 2.1 ;l are satisﬁed. Al- BY”): {(L+x1)+(L+x2)}+(L+x3) = {L+(xi+32)}+(L+x3) = L+(x1+x2+m3) 27 CH. 2: VECTOR SPACES and the same is true for (L+x1) + {(L+x2) + (L+x3)}. A2. Lis the zero element of JV ,3, since L+(L+x) = L+x for allcc GE. A3. The inverse element of L+x is L—x. M1. By (2), 1(L+.v)=L+lx=L+x. M2. a(ﬁ(L+x)) = a(L+;3x) = L+a(Ba:) = L+(aﬁ)x = (“13)(L+x) by (2)- D1- “{(L+=v)+(L+3/)} = «(L+(x+y)) = L+oc(=v+y) = (L+ax)+(L+ay) = a(L+x)+a(L+y) by (1) and (2). D2. (oc+,3)(L+:c) = L+(oc+ﬁ)x = L+ccx+Bx = a(L+x)+)B(L+x) by (l) and (2). Our question can therefore be answered positively. Deﬁnition 2. The vector space whose elements are the cosets of the subspace L of E and whose binary operations are deﬁned by (I) and (2) is called the ‘quotient space’ of E by L and is denoted by E /L. Theorem 6. Let L be a subspace of the vector space E and let D be a subspace of L. Then L/D is a subspace of E/D, and a coset D + a: e L/D if and only ifx e L. Proof. The space L/D consists of those cosets D+x for which a: e L and is therefore a subspace of E/D. Now, if D+x1=D+x where a: e L, then by Theorem 2, xl—x e D g L and hence x1 6 L. In view of Theorem 6 we can now associate with each subspace L lying between D and E (D g L g E) the subspace L/D of E/D, i.e., we have a mapping L —> L/D. (3) Theorem 7. The correspondence (3) deﬁnes a 1-1 mapping of the set of all subspaces of E which contain D onto the set of all subspaces of E/D. If D 5 L1 E L2, then Ll/D E L2/D and conversely. 28 I <Yfrji~ga-4— .1. .. A'it'. x.» - 2.5: COSETS AND QUOTIENT SPACES Proof. 1 . An arbitrary subspace Z of E /D is the image of the subspace L of E which is the union of all the cosets of D which are in E. Hence the mapping is onto E /D. 2. Suppose LllD E L2/D. If x1 6L1, then D+a71 ELI/D g Lg/D and, by Theorem 6, as] 6 L2. Hence L1 _C_ L2. It is easy to see the converse, that if L1 E L2 then L1/D S Lg/D. 3. Suppose L1/D=L2/D, then, by 2., L1 9 L2 and L2 9 L1 so that L1=L2 and the mapping is 1-1. . 1‘1 .. Av1=L+x V -’2=L+-”32 . .122, 4 3leer ‘ l i Fig. 5. Theorem 8. I f E is the sum of two subspaces, say, E = L1 + L2 and L1 n L2 = I) ”m E/D=L1/D 6-) L2/D (see Deﬁnition 2.4 ;4). Proof. 1. For each eoset D +1: 6 E/D there is a decomposition into D+x = D+(\$1+x2) = (D+x1)+(D+x2), Where 961 6 L1 and x2 6 L2. Hence D +3: 6 L1/D+L2/D. 2. If D+x e Ll/D ﬂ LZ/D, then a: 6 L1 n L2=D and therefore glzx=ﬂ Hence Ll/D n L2/D={D} consists only of the zero-element of 29 29. LeLW be a subspace of a vector space V over a ﬁeld F. For any 0 e V the set {0} + W = {v + w: weW} is called the case: of W containing v. It is customary to denote this coset by v + W rather than {u} + W. Prove the following: . Fl (2) v + W is a subspace of V if and only if v e W. (b) vI +W=v2 +Wifand only ifv, —-vzeW. Addition and scalar multiplication by elements of F can be deﬁned in the collection S = {v + W: 12 e V} of all cosets of W as follows: (”1+W)+(”2 +W)=(vr +v2)+W for all v1, v2 EV and a(v+W)=av+W forallveVandaeF. . . (c) Prove that the operations above are well—deﬁned; 1.e., show that if v,+W=v3+Wandv2+W=vg+W,then (vl +W)+,(vz+W)—-=(v', +W)+(v’2 +W) and (1(0, + W) = a(v’l + W) for all a e F. . (d) Prove that the set S is a vector space under the operations deﬁned above. This vector space is called the quotient space of V modulo W and is denoted by V/W. . -.-_-. --...—...v.-. The following exercise requires familiarity with Exercise 29 of Section 1.3. 26. Let W be a subspace of a ﬁnite-dimensional vector space V. Let {x,,x2,...,xk} be a basis for W, and extend this to a basis {x,,x2, . ..,x,‘,x,‘+ ,,...,x,,} for V. (a) Prove that {ka + W, x”; + W,...,x,, + W} is a basis for WW. (b) Derive a formula relating dim(V), dim(W), and dim(V/W). The following exercise requires familiarity with the deﬁnition of quotient space in Exercise 29 of Section 1.3. 30. Let V be a vector and let W be a subspace of V. Deﬁne the mapping n:V—>V/W by 11(0): 0 +W for vEV. (a) Prove that n is a linear transformation from V onto WW and that N0!) = W- (b) Suppose that V is ﬁnite»dimensional. Use part (a) and the dimension theorem to derive a formula relating dim(V), dim(W), and dim(V/W). (c) Read the proof of the dimension theorem. Compare the method of solving part (b) with the method of deriving the same result as outlined in Exercise 26 of Section 1.6. The following exercise requires familiarity with the deﬁnition of quotient space deﬁned in Exercise 29 of Section 1.3 and Exercise 30 of Section 2.1. 22. Let T: V —+ 2 be a linear transformation of a vector space V onto a vector space 2. Deﬁne the mapping 7': V/N(T) —i Z by T(v + N(T)) = T(v) for any coset v + N(T) in V/N(T). (a) Prove that 'T' is well-deﬁned; that is, prove that if v + N(T) = v' + N(T), then T(v) = T(v'). (b) Prove that "T" is linear. (c) Prove that T is an isomorphism. (d) Prove that the diagram shown in Figure 2.3 commutes; that is, prove that T = Tn. T V ————— +- z \\ I \ / / Figure 2.3 V/NiT) “l- vnuy. u wtuvvuuolhuuult 26_. Let T be a linear operator on a vector space V, and let W be a T-invariant _~ subspace of V. Deﬁne T: V/W —> V/W by T(v + W): T(v) + W for any 0 + W in V/W. (a) Show that .T is well-deﬁned. That is, show that "7(1) + W) = T(v' + W) whenever v + W = v‘ + W. (b) Prove that T is a linear operator on V/W. (e) Let n: V —+ V/W be the linear transformation as deﬁned in Exercise 30 of Section 2.1 by n(v) = v + W. Show that the diagram of Figure 5.5 is commutative, that is, "T = Tn. r V/W——-—*-V/W Figure 55 In Exercises 27 through 29, T is a linear operator on a ﬁnite-dimensional vector space V and W is a nontrivial T-invariant subspace. 27. Let f_(t), g(t), and h(t) be the characteristic polynomials of T, Tw, and T, respectively. Prove that f(t)=g(t)h(t). Hint: Extend a basis 7 = {3:1, x2,...,xk} for W to a basis ﬁ = {x,,x2,...,x,,} for V, show that a = {thr1 + W, . . .,x,, + W} is a basis for WW, and B B we 3:), where B. = [TL and B: = [71,. ...
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601-quotient-spaces - k 2.5 Cosets and Quotient Spaces a“...

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