Appunti ANALISI 1, matrici, sistemi lineari, spazi vettoriali

Algebra lineare

r iIIllt- :ìI liliilril1Nl Ln,ih/"\".UJ,rumvt til.irlZ Itrel.qli\urv'^)", , :I-r(;} rc[.r.hv. rn';- Xto"'iw\'i-\ %-R=, ^.() j &feu§-rh\nrì. alt'^' ZoI hqu.xÀt51ùr\nL"\r-c§-iomB:tlot. lrlril titrti-LY'= IitJ rlì,lfr it,x»+ lrir!,tl-/'L = .rtut»tUt4 /*ùavtetto#* .)Q^= ,h 3:l.: v?x,y E P--),l i t,(f) Kz€ l,ìr t!.1liill{lYllrK.r,i \l iX.Y*reK(V\ue,'C ?r.iuI -l7)( ulil{+ ittt IV,it, oÉq.--\=u'.t\-X, ìlli, \]\t i iR"i '.r',Pti* f.^esfloì- I [P*retrd-Q,tt§Yrytti-l-l-i-rr-( f*rrk ièÌt" liri'.ll Iil'l _j-i i . 1 Ill'lv'i lli i,

Struttura lineare di R

t/il l1\ ,iii :lIll.lri llpl.g llrl iriiir, lìx*yI S*,,1 LiR^@t\ÀtÙ-\a(tÙA ix l,Y+= .I g (Rh,1AS\,cogaq16 V li>r,,7) ir'ill lL,'l(L« $, eR*ÀEU{RoI e co Iùt'.ltl-Il:I t'lrI\ l.-*t oe&uru.rlo N\tll,\. ,0fu,ZÈrc/o\vo= [ \rl\o lv : ll/Y /-v, \r.\ l-*i-X=laQ,X= [ \o*Q-Qon.,.' - ( -x'"Xvr\ )) \/'/|L i4/ g IrG ( ol*' dfn'efQ,t.tlQqqggd.o{.t+r.VQi ur,rrr\0.ci^tht@ {{uZ PtJ,.(i»crr0s.nQUNIO{r.gL :t ì=(*,) lt: , ,Xe?-' I( *.\\x :\"'L 6r= \ \- §h\t:l >ù"/ :. ,PLTqelk(, I, *a +R"*,/ ,\r,r,",tttùt\:«-ttU5-§Vrr R-e,tftxea*+s Sa^t d{-eftr.r.r-. -RI/\^A(x*/ *,\x)A, IÀ.y= l$r"trurrpgào 5ccr+1u.'L, iI-\(A+H)x = l*Ax\x iI:lo VAr\oCtAttÙo i I,ì i,l iIlìl i''o 0h{xNUÒfectltV44 tt\e ì"irl i1ìl l, l:l Ir)ìtl l.,il 'ill lr '7 -t . ./rUsl.;.\?Ltuunnd)-.r.,*r 6a,6L1a Q " 1" fl*,4 ?rt+-giryrt -Rtt{or'ottocu) "t§àN,t4^4"rgA§$"r- c^,\fstr'1K"'cc)L\ cfeltc\fo.(),.a,q-' sQ^ LoLtlA u{tt&r@ CteLtfftrlC,teÀ :'l'l./\( O A)dc.ucrr..JZv )...JeL- ./-K-a>2e^' / 2\x.(4 ) Y= )\ . , L 4+L\ )=i

lrll lìil rl-ilil!iìI 1ji .-l- i -:--,--_lI t--r__.'f--_' Iii'_ . t1ii tji ilti II -i -'ì*---+- I----l -:--''--'r'- -t - r-iiiI itlil: i- l!:l !i rl -i- _l-i 'T'-I - .!l ri -1._.. i-i--iI iìi ll1tt Iì it:iii I j-r--iI:i --- --i',--T--'"i.,i-.,,',-- :r I_i_" 1_ i ri -'t----.. -i---itf_ Ia--i"irl,iìl Ittlil.-.'.- i..,-.*.,.'-','].rl i,,-_-.--_-_-r'_ i tr-.-. ..lriì i-*li- 1liji lìI I II) 7 -"I(2 ,'' 1 )-, i -At \L,2t= - q{-.4(\.'' -{(Éa tl'(t" Lt{-ui'\zt)'. ìaì.,/ ì(é1I ,'.|L'lrl:! lr/T__=*"__ -)ILJt y\rrnrr.o,!' X,'\,i ,,'./\ i,iriiiiìii Qroq:rQ X..ob.'Xo iilliiilt.c\ eQsrugrlJsoe ITurried Cgrri'qlv'aft €rr^soJrs -'-ic€€gP A]ot-ur€-,-, - -- <,l'i 6J'I , : I iiii"l"iiiiiii/11iiiiiliiiilit: lillitiiiiiI.l "td il0 *i itiì li1iI rilI ^=^l*Y liiiliilil IIli liI *r,I e't'tI Iii tor;uts' ,it t]I otI XsX<+O[.mearrrl I illtiì i I')l:l l:i ir:ilrll , ifc)l{urr1',0ÙU.\i i I iiil iii;Ì i r.i liiil ìI rli'iiriìlil iilillilil lllI;iirirl rll ltrtrIL

:ir,jl,\(3-Za,:iy =-LX,3-9,* f oè& ::' -() /\I O-o (otr,t .dtreAu, \..rArl\)ll -Zxqirr,u.,c,\r'A=3L lr l(\f +2/o, bf'*\ 3-le -2,,o. +-)o-,t/f/ e/l-i n: ,t*e,a:$ LNXeV TA»t?) = I)')-2alJq- r-- ar)-( I:O) Q= ,)( I.l \=r- -')\,Nr -)eÀ.Ìtl ;, \tllli r-3l/3\ \)(i)G,1! l, ?L)ri l ,\,/ l^=-Z ^ +)otl'q0^ r*,;0hp{,A§r.tc oUm Q.- o&[0',r-Qh0,o\^P.l/ ' P.,nl,rp:ÌIìI:ilgKnnol^.ùU Xn /rN.,À. Xz, ùtP'X.ltu,,r-s/Vrortl -. -t-e»J ,-' ':', t, I',I j'iI I ir, i ,\/ i l {!td^'e- OutJ" 9\d.r.'.rrrNo4,0&, "6crn1?t\nq1iiiii ,iiiiiiriiilc\l;i'flrtt , Ill,i.ii:lll,i,iilìllll I,iliiIILru»rùI,rft"[,loi ,;Xr,Xu.-,-iX'ÉL,t.--:,ii Illl/t I i,ii, --.- -l I, i i-a , i\-r I,i I , ì rI i Irtc.*$3t,p"\w) c-ombtu Aruua kdfi4r . noq,UÀr'{vEwe dÈ Oot^e t t,\cXeÀ,X. =Ql, \riz?&p4rcuuuru: 1OI Ài- hctotFaL "'oil^/Ld§,]\.\lv r-r.- &3orf \/"+ Xi-+tz\.rt À;.lIi t,tllio\ò. eRlX,yrtr)Ànrrc

lli_,.,1ltl_.,',iljl;rilll- iìiI*o..*.' --"-' {aA;J;1R'<L ./tN ùrP W'r$\>N eA*u.drL'gt^ &-t+,N, rrn o".&ò oAÉ ot[§-uùOu{rgvt& ÌUplhrtrdllw' rltth*,|Tlt ",I , V9^!/Ldr1l-L'q )l r\f,{\lrr, &qvUaeod* \$R\^Rq\< I ,lit, crEt\p4lstr(o\aI. tN§\l[\lqeNr4\^J "l ,1§ ò - (i Ix (x))* , 1,l;ri i llillxr Ilrp( rl(r), (0")l,2 I Bare?\1 ):( 9sU i,Q, erq-'1i,i ,i/1\ t o \{t \=e 'n't1, o l'' \[; )irli lY' \ t t.-.+9rX' erY, €wX^Ix* ,I,;l,ir x' M0L-t: n ,a\ n&x&",Xx dtXY, (-,l ::i lr ILi Il,l'ri

Struttura metrica

rtll'll'i|rl:lil'ili'iii lil,,rl rli'l,,,lllt'(prodotto scalare standard di R )".:t: ,:td (§) y=(t:,,) R:)(- e"À^c\\À"" -,- ,'__./ |txzy.< =,'/>x X.y, -*t\e» G&x,r/^r+L_ t il:ilil IY--€^. !()/\ \) I - i(u§^r.#aU tX ,./>, = ,1,")_ +,f_4\?_+rlttllllljltit : -.'., -lÌiii! X,Y +(x,y>\m'gc§ecù(Lrsr r0CInT ?</,x)(X,Y) .tI vxv€€*a.{r\.14 €(1"x *,.1#{'r^*1.**-n-À<\ tfitr,rr)t,v,ArLLrrf6tss§.skÀo iliriil j- ; Ii 1tr i ,. ì- ,iiìiitiilrii ilillirll .iiìi'rll il.t"-_,1_ iI i;1.' I' ;ii:t ii II : :, :i ':iirl - -I '-- -:-- -i----;- '"'i' -i-1"-| : | i , T' 'iìlillìilili ;iii;;i;iiii;i;i tl'ii, ilir ;:iri,r Ì ; :i,rirLi'ìii ,I,T-l iI..--r-.-f. , i; i rill,llii r i; I il- Lrììrl ,i|liii,lIl:r,-;iil I'lrl:r,-;i: i ;, ,i iiiI ii-lI ,| 'rl , iI; ' '' iilllliirirli,lirrlll

-, -i'";' - i'i..r l-i l ;--i, I II i , ,', - i \*i.it ]ty,l.x,i*,i,iIitrl \ Ioriii I i il.ir,1.0, ) ),)L=o X=Oitluu,Q,»o ,d 'iri l il»z x X=o<>iX )l ,t+iI 1I Irl llI I\-l i I), i rl4 I+ l tiI ,' i ,.ii i iI fI t'l'iI ir*'*i'i"iI Iiiiti l-,i iii,ii,: ,*r,^loru.i.Iiilli;;lo lii'i iiiitiirr,€Jt .,nr*ir+ X nùrrrg\.o IAOQ.§ fnogcr,.[r,UcrrbÀJr,f,#Ni.xtXrii.rr,: ,"il l1[ì, I1"i.il'IItllr l,ix/i iiiii,i iiitiiiiii, ,-t/, llllirilirl ;rllriliillllrr ',1:il Iiillliiiitii liil't'li itliiii iii i;I i ii ii i I I r:,lr:IIrIIìriIIiiiIiiriiiIìIiiili :iii:ilrlillirliiriiiillriliii'i;l

tt/ =)r\1 Jr'X, XurVutX,t/, .rXzlz ="-,il II !l+Z,La ,-+" I.=lL[r,) À.l I .= -/t' Ie<, v,il I\{ffi I,llx\\ ,l= j,j) i:#- ':tI .4, L\ 3" +L ;l'r,i iiI: L+à!Ir = ,t,l,i iillr llIll lt(--Iytt \.7,Y) -" Ii,IIlrrlìrI Itl,,J.§&. R-r'€X,&,(, Irl1lrìft I it II I Ll i.TI ! lli II tIIiiII lIlIIII iIlI II li lll I -ìì.--l i ili 1lII iII I lt;ii ,I) t,I ii II I ili iic lLLI i; it t :lI Ii L Iatlwil'l IIIiiIiII'-(l z- L: r"\ii iII 1li il riiI il1iIIt\Y\\'[y lt" ,lill{riI/ i>'ohN llr7ltli\X\itX ,r/ .iilli llir Ii,ii.I:il l! I?--tx,,tf L{ trytÉnXrÉ o\P- ' \I r./.tl /'tat^e J-/'l I t.X,\/>l: I ttXjr.-r(Vrt \Il/,xe,/ ilg$'r^=fup- J-']-Y .il rl 'Ox ) =o\À ../. /l '/l'lt :Aru-f-u. = Cc^orrU \a$JsbtrJ.\,^ek-=oul,\ttruor 91! .)iii yi'1r \---iii )iit iit §, \=otW,.ùn:[,- Iq o,y, e,Kn,5ì"* ,]i\\x- v\l LI'I lIrl Lil II,(n IIrlt:lll

dad[tu]&ct ÉU ccr \)EA Llì :x., *^-{ \a*+-]= R.A-oL(&5 x'r.,Src,r ur'Yrc,t- V""tr*.\T L.i\r,v twfu ,4!,r'€riq)o.,*flInr6-",r.\q J^ 63V L Ct,so-cxL :,crrktrgotrcr-Ql o\ÈCI-u-oi .t1X,,Yi>=o !i*j,Y/\r, ./L ]-J,-Lr',lfl/ wrt l,1/*rXe)er\ ,1lrlljlliil,i,liì1r.ri ,L ,lL1 1l IlI'rll ìiili;t,6:dP lrpntdlJ&Lo-AQUthaÀe<c"ae, it:ifuqP?'\'ltsff,rirt,latp+l ,Lrr.,, t;rl,llr.Iii ii ii Iilìri,l in^".nt-l"Jlrli ';,i,ii:llIt,,:ll ./' ^ t,/' r./-\ LfJ(rro'- ,/\t rl \ -,?U\ '\ '-'\\- ."' 1\\*t:\J'l i,{l t!-'i!-' .j,"ìti 't---tt ltrr)€q-.\n"q Pot*-?.a'r\n"gf-rOnI rrwLriklrur'(JLti'l-qirt =\.",X)it[ll XctL' \sv .*u{re/L2 §Tbtse+"*qo% ìC +)---'u( *§;L= r tilri:,i\,)- IiJ iiì-i-ilirI Iili .'ili i-i*rtl/t le ùrv\oL +NlA

ltiiilliì1r-: Ij-"i I--1- I |i l--ii- Iili .i I iI !I ìI.,Éi fI I i i i il-t',1" i--t---Ii-- -+--:- 'f__ --i'iijI l- )i-i iIli I i IiII I 1-..I lIìr: li 'a'i.' : ;'';''I Il '|'ItlI i l III Ii1ì I l ì.:i i ,i1_ rlI I ì"l- ':i- -1 _ i'.-a_;-- t-i ii :ll I I II I I:.": l-fi iiiÌi l iI I I--:-I--. ! i-ìI ,i i--.-l--- .i _,1-,:.--. II T'tI l-lili t: LIirI Ii II I II il! Ii I'i..-'i_ iI i'.i- -i'.l,-. I j,,,;:-. . i_i;tt I III Il LI -t--- l-,-i-r Il -t'I1 -ì 1'-iii I II I -i- -i'-.ì*l I II -i i------I ì IL--l Ì i'--i-'iII Il il_II *i'*, l 1iiil+lo-i Iit I :,iX;.>X,iillliii iitltl I4'+o'= =)t\-/lli.i''i],:iiriirilì,.' a,, 4?=Ot+= iiitì'lllillliiri,iilil ,,4n* qvrugl-rt:rlee:Jt_rlh 'i.' : "^"U1l. =g-,}!i II i' i I I I I d'tÀrrrslrst4}r1"' riiì,l tlttIIrill\r' /o ,l(t )X=lJ-u\. 1=[LT ÉiK\wVaud.d'gr.^- Y)=2g\,^"'. +.,e)jy,x/ \?,' IIL«rlyeÀu-q-, :( Àt ryt'r" =tc :1\,1 :o( \:t,r^l :o\-,t"1(tIU M"t"'»zwgaoàqZu-&W. 1*\afu=^sr')M Il,iilll lil.ltttri t ;Wd'"doirgtlt 1-.+{+d.ll,"o