= y and Normed Linear Spaces

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1 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads statg wth the last ow The soluto ca be smplfed b ealzg that the compoets of coespodg to the colums of à that do ot have a leadg et ca be selected abtal Eample: 0 Let = 0 0 be the echelo fom of some mat We wat to fd N{ } = N{ } ξ ξ 0 ξ3 = ξ4 ξ 5 We have ak{ } =, ullt{ } = 3 The compoets of ξ coespodg to the colums wth c dces K ae ξ 3, ξ 4, ξ 5, ad ca be selected abtal To fd thee leal depedet solutos that spa N { }, we set ξ 3, ξ 4, ξ 5 to the caocal bass vectos of emag two compoets: 3 ξ ξ ξ 3 ξ ξ ξ 3 =, = 0, = R, ad solve fo the ) ) = 0 : ξ + ξ = 0, ξ = 0 ξ = 0, ξ = 0 = 0: ξ + ξ =, ξ = ξ =, ξ = L8- /9

2 LINER SYSTEMS 3) = 0: ξ + ξ =, ξ = 3 ξ = 0, ξ = Theefoe: N = spa, 0, The No-Homogeeous Equato = Hee, ae gve ad s the ukow Sce we have the equvalet homogeeous equato = = 0, [ ] = 0 o: W = 0 W, Thus, f m F, the the augmeted mat W s m ( ) + The solutos a subset of N { W} whch s a coset, eg, a subspace plus a costat vecto To chaacteze the coset, let deote the estcto of to oe of ts suppot spaces Ssp ( ), sa, D{ } / N { }, ad to ts age The thee s a soluto of D / N gve b: le N { W}, but the last compoet s costaed to be - Theefoe, the le = fo Net, eve soluto of = = must le the coset: + N L8- /9

3 LINER SYSTEMS Poposto: Thee ests (oe o moe) solutos of = ( a) R ( b) ak = ak W Poof: (a) Ths s clea (b) = s leal depedet o colums of umbe of depedet colums of = umbe of depedet colums of [ ] { } {[ ] } { W} ak = ak = ak Suppose s a soluto of = Choose a suppot spaces ( ) Decompose the vecto as: D = U = +, S { }, N{ } sp S, sa, D{ } / N { } sp C = V N{ } R c θ U θ V N { } c R L8-3/9

4 LINER SYSTEMS Sce s vetble, we have = + = = =, ad: = 0, ae two solutos: If Hece, eve soluto s of the fom: Cocluso: Eve soluto of = + =, = ( ) = 0 N + N = les the coset + N { } N + N Poposto: The umbe of depedet solutos of { = s gve b: dm N = ak{ } = ak{ W} L8-4/9

5 LINER SYSTEMS 3 Nomed Lea Spaces The om o a lea space s a atual geealzato of the legth of a vecto o the Eucldea space E ove the feld R, eg, the lea space ( R, ) 3 Ie-Poduct, Nom, Metc Spaces 3 Defto: Ie Poduct Space R o whch = = e poduct space s a vecto ( VF, ) o whch s defed a fucto, : V V F such that the followg aoms ae satsfed z,, V, α F : (), =, f F = R o, =, f F = C (), + z =, + z, (), α = α, (v), 0 ad, = 0 = θ The om of V assocated wth a e poduct space s gve b: : =, 3 Defto: Nomed Lea Space omed lea space s a odeed pa ( V, ) whee ( VF, ) s a vecto space o whch s defed a fucto : V F such that the followg aoms ae satsfed z,, V, α F : () 0; = 0 = θ () α = α (subleat) () + + (tagle equalt) L8-5/9

6 LINER SYSTEMS If we cosde fo stace ( V = R, F = ) R o whch : = =, the: () () () states that ol the zeo vecto has zeo legth ad eve othe vecto has postve legth, states that f a vecto s multpled b a eal scala, the the legth of the vecto gets multpled b the absolute value of the scala, states that the legth of the sum of two vectos s o lage tha the sum of the legths, eg, + Eamples: (a) ( R, ), : = ma whch s the l -om o R T T Check (): Let [ ], [ ] = = R The, + +, =,, (tagle equalt of eal umbes) ( ) + = ma + ma + ma + ma = + (b) (c) (, ), : = R whch s the l -om o = (, ), : = p p = p R R whch s the l p -om o R, p [, + ] The l p -oms tpcall used ae the l -om, the l -om, ad the l -om (d) p ( C, ), : = p p whch s the l p -om o = C, p [, + ] L8-6/9

7 LINER SYSTEMS Notes: Both R ad C ae eamples of fte-dmesoal lea vecto spaces, ad cosequetl t ca be show that gve a two oms, o R thee est costats k, k such that: fo eample: ad, a b k k, R (o C ) a b a, R,, R 33 Metc o V ad Covegece We ca ow use the e poduct to defe a topolog, eg, a oto of covegece V Thus, we ca thk of the quatt as the dstace betwee vectos ad, defed b: d(,): V V R+ d(, ): =,, V Ths s called a metc o space V The popetes of ths metc follow fom popetes of the e poduct ad the tagle equalt o V : () d(, ) = d(, ), () d(, ) = 0 =, () d(, z) d(, ) + d(, z) Defto: Covegece ( V, ) = sequece of vecto { } tege N( ε ) such that: ( V, ) s sad to covege to 0 V f fo eve ε> 0, a d(, ) = < ε, N( ε) 0 0 That s, the dstace betwee the th vecto ad ts lmt teds to zeo as teds to ft L8-7/9

8 LINER SYSTEMS Note: I ode to test the covegece of a gve sequece of vectos, we eed to kow ts lmt, whch s ofte dffcult to have (ad s kd of a chcke ad egg poblem) Theefoe, we eed a test fo covegece wthout the kowledge of 0 Ths s povded b the cocept of a Cauch sequece Defto: Cauch Sequece = sequece { } N( ε ) such that: ( V, ) s sad to be a Cauch sequece f fo eve ε > 0, a tege d(, ) = < ε, k, N( ε) k k Thus, a sequece s coveget f ts tems appoach abtal closel a fed elemet, wheeas a sequece s Cauch s ts tems appoach each othe abtal closel Lemma: Eve coveget sequece ( V, ) s a Cauch sequece Poof: = Let { } be a coveget sequece ( V, ) wth lmt 0 To show that the sequece s also ε Cauch, suppose ε> 0 s gve, ad pck a tege ( ) The, b the tagle equalt, Hece, sequece { } ε ε j j < +,, j N( ε) = s Cauch N ε such that 0 <, N( ε ) = The above lemma shows that f the elemets of a sequece { } ( V, ) ae gettg close ad close to a fed elemet, the the must at the same tme be gettg close ad close to each othe The et questo addesses the covegece of a Cauch sequece Questo: If the vectos ae gettg close ad close to each othe, ae the also all gettg close to a fed vecto ( V, )? swe: Not geeal, ufotuatel Hee s a couteeample to Cauch sequece coveget sequece L8-8/9

9 LINER SYSTEMS Cosde ( V, ) wth V := space of all polomals defed o [0,] wth eal coeffcets, ad : ma ( t) = t [0,] Let = k= 0 :[0,], ( t): k t k! V be the th vecto the sequece 0 = (as a eecse, show that ths sequece s Cauch) We ca see that thee ests a fucto whch s, the om, the lmt of ths sequece, amel the t epoetal e, howeve e () V as t s ot a polomal That s, the sequece coveges to a lmt outsde of the space L8-9/9