crcis(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdftoem e flx)fa idd ma, 02010 ten...

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-( Toem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El) (, 3 . -. 6e)as a ns m in E F( hag raa& e) an d -a () oun E tQ =O ta E. we uia: hits ahell ane d ae(«) ¬ F(]c£( en lanye p ssi () Tdu ate Cohcass rm F F(x) isaon sde,4 FCD E FL ) sk o- E F. Ekap Pm)= at is eceaducsbe ia R E =R(/20a keld, is a fa4 R** n E (L-A) ( *L) wsual ke d ,E ,E Ct CRCiS(i) = (xt i)r-i)

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Page 1: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

-(

Toem e Flx)Fa idd MA, 02010

Ten FC*\/(LD is a Ad isTaduiye

Rann FcE

Suppo sredh, & F E = FL1/(ec)

a a vaa e n t ta

El) (, 3 .

-.

6e)as a ns m in

E F( hag raa&

e) an d -a () oun E tQ =O ta E.

we uia: hits ahell ane d ae(«) ¬ F(]c£(

en lanye p ssi () Tdu ate Cohcass rm

F F(x) isaon sde,4

FCD E FL ) sk o- E F.

Ekap Pm)= at is eceaducsbe ia R

E =R(/20a keld, is a fa4 R** n E

(L-A) ( *L)

wsual ke d ,E ,E Ct

CRCiS(i) = (xt i)r-i)

Page 2: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

2

Era-la oen2 (o dynu 2

Cre du cioe pono

On ft

xT« = (xt4) (x*Ltt) oLe

da4(mod 2) 2ats ,d+ In ty

no vas F2 dRes vai adss ,

*,,te. s aoid onfusb

Le eFI«] he nnons to.d paoial. Tu

exiss J& E Con F uoh Naf as faotio E.

wona

Pra Tala ae euck fa p(e) P(x)

E Flpu) is a i , F>F (wle ss dp=1) inFnfi f.

ois a ( p(an a (awt R(a) in E

- l p ()(x-4)8 («) 1

kas Colicirake tn F,

Tu s e F(z) haonon stand poly no naiad xsts

Ranso E Canseiai F suhk Oak Pakas h

ahens in ,

) C(z-d) (x- 42). . (-d)

Page 3: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

S

P ucdon on & Rve dy fastors in

Su h =n. RlE, F (ww E and n Cast pr

e() (z- 4,)3(x)

APp In dch sp do dE,1ns6ca lF) ano

LALams

nE, fackons ( c (x4).. (-4)

P (c)= c(7-4) (x-4).. (-)

Page 4: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

-4 FEE RKI exfersi on

Fe P(x), FeF[*ea , ncussa fra du che ode had an tan Fas a sußH

FeR

P stuc sR is an a hiian Op un di addban

Can {

F a k as scalas

Dh F-veuon s a Can vecls spaca on F sadue A

(V, +,) ne (V,* an a an na (vehs undlu

aids ,an FxV V (sca lan

ar)a,v) a,v) a

Bra,Le PveV a(Lr) = («L) r

) be P yeV a)v= aVt bv 3) s e Y,we o (Vt u )= aV +as

4) 1:=V ve VN

Exans) Glan ve dony F* , av

a

)Oeden s poa V{o lo.4ha

Ov O VveV (-) V= -V ve V Ax

veke suspae WcV as sud on W(V, t)

weW for a eFw EW

Page 5: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

S eV A ulst RVis cale -S s,. SGS,dibred, Va,

a

uon uladars an nes S, exoapt OsrOs,. Os,= S as e l()

bna naba Le maAs B. S w

I). e - (?) )is abtt F

Vis can dina i has B {V

Thn Drson oa ve ckon s(a is ue(-Gnd

Vhe - da mers on has a hite ha si

Al ha c«s {V hale (u Sare cadhai ( prasf /k (ane

R 07 Ovn

V &n V

KAs, Sa esppaialk 0ak au elpenras aad eig meda)

Page 6: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

FcFC*)

Pco FCis a vcna ~( hasis , ?,*, ..

awaA Ff*} has a unua

Ke)= at9,1 Br S ", d, a

Fl/es is an F-ve k Spaca ma a a s 31,,.. c

wwwq sueh puerkah-e e

u q

S v; s ma

Can al&

spa >,x,. x*"is a hasis

Car enceda r b Caheleds (,.. - D

Spe ca ca:is ircaduuci ola E F Ry sa A Spe ca

E is a P-veun> paca unorson af m hatis

te u a Re a E=Fl0/tiA), besis i,, ,"}

Page 7: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

FcE AS ekrSian E is an ve e n spal

Re -Amonal/F Extensn is cale te

naose n E as F-vee paca i cao u coR extru Son

E leF) oon

Kampks E F CF: FJ edensun ra

si* ),a} Sa tbi,e R R') 2) R CC

3 eR ie exteason (Qis nu, R s un cou da k)

4) QcQLE) aebi l4,ebasis af$] is (t,E).

ta -2

S)FePC»]e F(z)=3 { 9,eF ]t equiv. uakon

FCF(C) hork dra

) h-6Atn) )basis LF -2

P) iy :)-3.

Page 8: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

-8 Pr A e SIE has o Pcop

E hos ohana chai s6t p, r Se O P

CE F E isa een s2ca /a

3asis (e..e &E/E

u an e Eas a wa prey-taton ae,ra21 to,f a F P claias B- e<ck . .

For eac po an a> fu e xits a uo p d TLnen

P nA Sonep n , RA

noahlon.

F,F

F no a sueA.&iso

F

eFs, )F wson

Page 9: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

EcR Comm&ae w chonsists p

(ahy Z (e)a'L. a'- (f)a*6

Prog Cl,2,.P P Pd, does ne d

2. Pe-1).. Canal at rom

il-). (p-i (p-i--1).. 1

()*o t, ()-o nF il,2,P1

Gul (aeb)a r ,eR as ahowR

P (a-aL

Nok (a.) = (a «) = latt") -a'6. an ke

Ca e)Pa Rr ,bhe R, cR, l xenuE

H pl (),2... -()-op.

s labP= aPLP adhe arabloa

(Aa Cm tadme un G O map aaf r aeR, FcR RR (en Ionenpis is mpusm

kom an o6^d isell) Co robendas e nsnapuisn

Fr( a? Rr: R-R Rr reesks addikon, maltlicsd

Page 10: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

Fr()=a OO

Cxenu wonk o f a edon

ta e c8r 07ot p-i a

(Feat 4e (usn) al=a Ya ef (adol O).

it Poroi (zJ= *P-x kas p di'sb'nck taols p. foclos

- x (*-)(x-2).. (x-p-i x- * -2). . (x - (p-0)

PS *-*(x-)(x-3) (x-Mx) P-3 - =z(z-1x-2

-e c(x *r)(* + 4+) (x+ d*t)

FC3

n n) g F F, F s endu qp (F isan Feu pac

FC ae F n F.

-

Page 11: CRCiS(i) - math.columbia.edukhovanov/ma2_fall/files/lect_8.pdfToem e Flx)Fa idd MA, 02010 Ten FC*\/(LD is a Ad isTaduiye Rann FcE Suppo sredh, & F E = FL1/(ec) a a vaa e n t ta El)

l exisls a rte exttasun we pcs

focdas in E.

k Fac El at-a'l Sustall k *l-

FeE is a SkRRI,

P s,beFalea, b'- b) laJP=t'art (abaPs iF.

Fa suluns {E

aeF, ato » ()=a" a)-a alef. Fis a sus hA -l F.

F

nE(a. in P aa-)=q Tb e otTF=q ado sla Ot - hes no

eras uss.

A () hasa cepad taot (uulki e « nod)

(-4 () (x-(x-+3te) has a cpea ca, (ns o R o1 C)

'(z)= (x-A() *2(zwg ) (z) ((x-)g'(<)12 g(x)

-s| ?(E).

( s a nuhkr taot

F ()=nT*, ner.

()=q (z)- - (-x,-)=1. ae liskac