Finite Dimensional Vector Spaces.
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1 Lctur 5. Ft Dmsoal Vctor Spacs. To b rad to th musc of th group Spac by D.Maruay DEFINITION OF A LINEAR SPACE Dfto: a vctor spac s a st R togthr wth a oprato calld vctor addto ad aothr oprato calld scalar multplcato (a rul for multplcato a ral umbr tms a lmt of R to obta a scod lmt of R) o whch th followg ght proprts hold (aoms of lar spac): () + y y + (Commutatv Law of Addto) () + ( y + z) ( + y) + z (Assocatv Law of Addto) (3) Thr s a lmt 0 of R so that for vry R 0 of R: R (Estc of Idtty Elmt for Addto). (4) Thr s a lmt y of R so that y y for vry R y of R: y y R (Estc of Addtv Ivrs).
2 DEFINITION OF A LINEAR SPACE (cotud) (5) α ( + y) α + αy. (6) ( α + β ) α + β (Dstrbutv Laws for Scalar Multplcato ovr Addto). (7) ( αβ ) α( β ) (Assocatvty of Scalar Multplcato) (8) (Idtty Elmt for Scalar Multplcato) 3 ADDITIVE IDENTITY ELEMENT ( ZERO) ELEMENT Thorm Th addtv dtty lmt s uqu Proof. Lt 0 ad 0 - ar two dffrt dtty lmts. Usg aom (3) w gt Thorm For ay vctor R 0 0 Prov ths yourslf. 4
3 ADDITIVE INVERSE ( OPPOSITE ELEMENT) Thorm Th addtv vrs for ach vctor s uqu Proof. Th stc s guaratd by th aom (4). Lt for R thr ar two addtv vrs: y ad y. Th w hav y y + 0 y + ( + y ) ( y + + y 0 + y y ) Th addtv vrs w usually dot by Thorm For ay vctor Prov ths yourslf. R ts addtv vrs s ( ). Thr s o such oprato as multplcato or dvso of o vctor bу aothr for gral vctor spacs. 5 LINEAR DEPENDENCE AND LINEAR INDEPENDENCE Dfto. Th vctors,,..., from R ar larly dpdt f thr st scalars λ, λ,..., λ, ot all zro, such that λ λ λ Dfto. Th vctors,,..., from R ar larly dpdt f quato λ + λ λ 0 has oly o soluto λ, λ 0,..., λ 0. 0 Both dftos ar ust th sam as prvously. So all proprts of larly dpdt ad larly dpdt vctors hold cludg Ma Lmma Ma Lmma Lt ach vctor of a, a,..., aca b prssd as a lar combato of vctors b, b,..., bm whr >m. Th vctors a, a,..., ar larly dpdt. a 6
4 THEOREM OF AN ALIEN Thorm of a 'al' vctor. Lt a, a,..., a ar larly dpdt ad a, a,..., a, b ar larly dpdt. Th th vctor b s a lar combato of th vctors a, a,...,. a Proof. If a, a,..., a ar larly dpdt th from λ a + λa λa 0 follows λ λ... λ 0. Lt a, a,..., a, b ar larly dpdt. Th thr ar λ, λ,..., λ, λ +, ot all zro, such that λ a + λa λ a + λ +b 0. If λ + 0, th λ a + λa λa 0, whr ot all scalars ar zro, whch s mpossbl. Th λ + 0. So λ + b λa λa... λa ad λ λ λ b a a... a λ λ λ BASIS AND COORDINATES Dfto: a systm of larly dpdt vctors,,..., s a bass of a vctor spac R f ay vctor of R ca b rprstd as a lar combato of th bass vctors ξ + ξ ξ. Th umbrs ξ, ξ,..., ξ ar calld coordats of th vctor th bass,,...,. 8
5 COORDINATES ARE UNIQUE Thorm. For ach bass th coordats of th vctor R ar uqu. Proof. Lt ξ, ξ,..., ξ ad η, η,..., η ar two dffrt sts of coordats of a vctor R: ξ + ξ ξ η + η η If w subtract o from th othr w gt ξ η ) + ( ξ η ) 0 ( ( ξ η ) Bg a bass th vctors,,..., ar larly dpdt. So ( ξ η) ( ξ η )... ( ξ ) 0 η Ths mas that ξ η, ξ η,, ξ η 9 WORKING WITH COORDINATES Thorm. To fd th coordats of th sum of two vctors o should to sum up thr coordats. To fd th coordats of th scalar multpl of a vctor o should to multply thr coordats by ths scalar. Proof. Lt ξ + ξ ξ ad η + η Th + y ( ξ + ξ ξ ) + ( η + η η ) ( ξ + η ) + ( ξ + η ) ( ξ + η ) y η. ad λ λ( ξ + ξ ξ ) λξ + λξ λξ ) 0
6 DIMENSION OF A LINEAR SPACE Dfto Th dmso of a lar spac R s qual to ( dm( R ) ) f thr ar larly dpdt vctors,,..., ad ay + vctors f, f,..., f, f + R ar larly dpdt. Thorm If R has a bass,,..., th R ) dm(. Proof. If,,..., s a bass R, th th vctors,,..., ar larly dpdt. Th ach of th vctors f, f,..., f, f + ca b rprstd as a lar combato of th bass vctors,,...,. By th ma lmma th vctors f, f,..., f, f + ar larly dpdt. BASIS IN A LINEAR SPACE OF DIMENSION Thorm If dm( R ), th ay larly dpdt vctors,,..., ar a bass R. Proof. By th dfto of a dmso thr ar larly dpdt vctors R:,,...,. Lt b ay vctor of R. Th by dfto of th dmso of a spac th vctors,,...,, ar larly dpdt. By th Thorm of al vctor th vctor s a lar combato of th vctors,,...,, so,,..., s a bass of R.
7 SUBSPACE Dfto. A subst L of R s calld a subspac of R f for ay L, y L th sum of th vctors also blogs to L: + y L, ad for ay scalar λ th product s also blogs to L: λ L. Thorm. L R s a subspac of R f L, y L α + β y L for ay scalars α, β. It follows from dfto. Thorm. A subspac L R s a lar spac. Proof. Aoms ()-() ad (5)-(8) hold L bcaus thy hold R. Lt us prov (3) ad (4). Lt λ 0, th λ 0 0, so L has a dtty lmt for addto. Lt λ, th λ ( ), so L has a addtv vrs. 3 DIMENSION AND BASIS OF A SUBSPACE Thorm. If L s a subspac of R ( L R) th dm( L) dm( R). Proof. Larly dpdt vctors R rma larly dpdt L. Thorm. Ay bass of a subspac L R ca b tdd to th bass of R. Prov ths yourslf. (Ht: us Thorm of al ). 4
8 SPAN Dfto A spa of th vctors,,..., s a st L(,,..., ), cosstg of all lar combatos c + c c : L(,,..., ) { c + c c } Thorm. A spa of th vctors,,..., R s a subspac of R. Proof. Prov ths yourslf. Ht: chc two codtos of a subspac. Thorm. Th dmso of a spa L(,,..., ) s a ra of th systm of vctors,,...,. Proof. Lt ra(,,..., ) r. Th,,..., thr ar r bass vctors,,..., r. Thy ar also bass L(,,..., ), bcaus thy ar larly dpdt ad all vctors from,,..., ar lar combatos of,,..., r. Th all vctors of L(,,..., ) (cosstg of lar combatos c + c c ) ar lar combatos of,,..., r. 5 A SUM AND INTERSECTION OF TWO SUBSPACES Dfto A sum of two subspacs L R ad L R s a st L + L, cosstg of all lmts such that +, whr L, L. Thorm. A sum of two subspacs of R s aga a subspac of R. Proof. Prov ths yourslf. Ht: chc two codtos of a subspac. Dfto A trscto of two subspacs L R ad L R s a st L L, cosstg of all lmts such that L, L. Thorm. A trscto of two subspacs of R s aga a subspac of R. Proof. Prov ths yourslf. Ht: chc two codtos of a subspac. 6
9 A SUBSET OF ALL SOLUTIONS TO THE SYSTEM OF LINEAR EQUATIONS Thorm Th soluto to th homogous systm of lar quatos uows s a subspac of R ( R cossts of all vctors of th sz ). Proof. Lt,,..., ad,,..., ar two solutos to th homogous systm of lar quatos a + a a 0 a + a a 0... am + am am 0 It mas that,,..., ad,,..., mas ths quatos bg tru dtts. 7 A SUBSET OF ALL SOLUTIONS (cotd) But th lar combato of th solutos α + β, α + β..., α + β also mas ths quatos bg tru dtts: a( α + β ) + a( α + β ) a ( α + β ) a( α + β ) + a( α + β ) a ( α + β )... a ( + ) + ( + ) ( + m α β am α β am α β ) α( a + a a ) + β( a + a a ) α( a... α( a m + a + a m a a ) + β( a m ) + β( a m + a + a m a a ) m )
10 DIMENSION OF A SUBSPACE OF ALL SOLUTIONS Thorm Th dmso of th subspac of soluto to th homogous systm of lar quatos uows s qual to r, whr r s a ra of a systm. Proof. It follows from th Thorm o th umbr of fr varabls. 9 TRANSITION MATRIX Dfto Lt,,..., s a tal bass ad f, f,..., f s som w bass. A trasto matr P f from bass,,..., to bass f, f,..., f s a matr p p... p p p... p P f p p... p such that f p + p +... p or a short form f f p p p p +... p f p +... p 0
11 INVERSE TRANSITION By aalogy w ca costruct a trasto matr Qf of th vrs trasto (from th bass f, f,..., f to bass,,..., ), usg th rul q f W wll s th t lctur that Q P. f f CHANGING COORDINATES OF A VECTOR Thorm Lt vctor has coordats a bass,,..., ad f f + f f a bass f, f,..., f: Problm short W ow,,..., ), Fd,,..., ) ( (
12 DERIVING A TRANSITION LAW f q f ( Usg formula for vrs trasto f Comparg coffcts by f w gt or matr form or q f w gt f Q f P q q ) f W gt from hr or full form P f p h 3 BIBLIOGRAPHY. Carl P. Smo, Lawrc Blum. Mathmatcs for Ecoomsts. W.W.Norto&Compay. Nw-Yor, Lodo Chaptr 7.. Stph Adrll, Davd Hacr. Elmtary lar algbra. Harcourt Acadmc Prss Chaptr 4. 4
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