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CA la faltan z 0 2 y 20 AN coy 2 TECBNGULaA.r A Ñ SUMMA M SN Colon, Afogorad AO aguero UPEen — triogul ac gue eS ER MA ok Ax Al WE ria ES « PR0BuUCT CF MATRÁXMOS A is A SINGULAR = mon uroix > (AL=0 AA | | | | | X-h is NON CONFORNABLE SAS Qs ] 3x0 PoAdas iz dimensions ase ¡acople DEF: Given Annan; BAT iz A is NON- SINGULAR, WarNx de. lA1 30 O E at adi = [2 IS n?* ESACN” ha (Ad = ma : : a LAS Ana >>> Ona * CRMMER WULE sn. 44 14 2.44 14 Mr yo. =4 Mas a 1.4 x + y+-r=4 ES Int Ñ lar tn y *e2=2 au s Gita AX=B) to firma X ur proteed Ey «Cheox iz E =0 A MON — Singular wsrma 7 «iq 205 > use CRAMER MULE , 4 MU yA) => NO SOLUTVON ar PARAMETRIC SOLUTON VECTOR, SPACES Tie Sex pgoverrorS Di, no Ot lintarty depor ¡E A Ka Ma kn Y ka o + En cOn = 0 SPRNS ÁM passilae Únear combiralbions og He verors ¡nm ae span - BDAMS To Segine a basis in Rowe need n Lincarty indeLgendeni metors + SPAN CHANGE Fina (eE mspece to 0 basis 24,0; 4.40> M0 «Errl) a a B= Az LINEAR MAP OR AÍNEAR TRANS FORMATION deb N 0má N' be bo veros spam aer kk. hlincar map Lis 0 funcion Assigniag 0 Unique Vector f0) in y! to eomn x ia Y A . PROPERTIES E ¿0 - 3 ¿en = e] Lineor was convenio senor SUbSpaces in Y inro Vector sulbspacis TA Yo, QUADRATC FOAM Y RU> Ri OO RCA 060 3xX.A0x Aomust bea UA Len nba >i.a Ex: SN EN ) 20 2 o > Olxyi) = SS xy = xo ay) 4 39? + 2xy + Una 220) [4 na3/lí QUO) =P. Asx = o; Qijo Xi? Lor 34 Two wONS +0 de Jinr a S- gor » Take + assouarted warnx +. Take a pomawal al x.41% where cock derua is monowiad pl Ea degue. Ex: Give Ye pAguoial expression Jo Q- form described Hirough. : Aa Ah 4 O (xq 2) ES Amr bo o 1.0.2 kotolke into account main diagonal am cocpf!s ol sopa terms Give we ossociared Wabris to Mis pay AOiAS Qlrqad = OS | + 3x2 + Ey» Az /f-1 2 312 20h su. 2 Apgli tabions o Q- Foru * Swéy he siga ot Q - Forms OLÉ is soid to te POS[TIVE DEFINTE (PD) S NTFÓ, ORO (E is ua to SEMiPOSíTiVE DEFNITE NRO, AR) 20 O (xd is sud to be NEGATIVE eridiTe O YX 79, A(x1< O a a (x) IN PRACROE po eetovarons 2. Use al EIGENNALVES 2 m>0 Nit 2 = Positive Deginite. 2. M0 ana some As 70 > Sémigos- deginite 4 M0, Vic 2 negar define - PS » vee ay principal minos Az /Qa Gm Os Aa = 1Qwl= Qs da > [00 a Ara Gin ls Az = am ar as Qu An Qs Mr Os am An am Qi Daz As], 1Bx1, 1631 >0 P.D- It, 121 >0, m3] =0 5.?.D. 1441 20, Im] >0, Va 120... minors change Signs starting wide mega, N-D. Examele : he fr 2 3 16.1 = 90 IS ES UNTEFIN(TE sh 2 po Iasi=p-4 2 sa] >0 2.0 h SIMULATON FOR THE ExAH. Assome 0 fincas map debio MES «Fiad 3wogc Memel ama A bass for 4.23 He ema. “13 A arogowali role 7 + 3s A sguaacine ? 1850, Oosarty de associoled O au 2...» tio. Ya + Lo -.- 3 » Kes (p= 1DEn/ ¿0% - 30 +40 +2DLTO A.3:0 sa y/0y zol 2 130. t209+408=0 =103=0 ker (añ 2 404 l 200 =) 20 + 203530 w,= 9 4x3 Jos 4 + Lu +3b4 30 020 ¿Checix ok Ju Lay = Span og He comas 00 A 2 pr fa) Y e arrage a SP 1(8,2.0,(2.011),(4,2,3)1 7 805: zo 2 (51 | UML Ax + Ze 42123 k “a ua = Ur 2 + 32 » lazAra = [3 24 4 SAA AMA A 2 pee n= lA l 3-A " Ker (A+ 18) = kec 4 A q RA pe Diva (her Car sad) 2 Mg AY AR os LA) pp. Yr2E=0 > sa 14, 2.09, (0,4120 % la (3d = A = ua 3) 3 4 diosa le DEF: Lear He gradiente bel a guncrion ar o port, derivabives ato goint Yo 10) = 3] (xo) d (xo) TC Xar 20 1 an Ex: gos = 154 4 > Io. ar ” MA 9 e > Es CERO DEF: e OM te Taro Nori, 3 4 0x0), ta y He worsx wr w gradientes AS Tous UCUorS, wnes As NACO) le He vector e paras E USES OF THE IMCOBIAN . computing disectionol AENVANVes Ji (a Ce) 34 0%) Ex 40040 > lay tax) Fina TELA) Ss 2 ¡CIS pig ) E dto x de IYLLO, Up h dl ap Ys, E AS dy 2 ca el 8 A 2 (9 = EA Y = énC y q SN y Properties oh He qradient A folrd z VE prR—R e hntar O.Qprorxi motion 0% MUUYamnodt. fonctions Flota) TFLO) + 3 (a) he Do Aimension 2 s 2 yl a = 3 2. Gradient Wauor Shows tHe gotas ACASO (iMUNOn ar point ka. Ex: gy = e > (4,2) — Try +0 qe Lo (1,4) = 3 a - 4 3 Y = (24,23 = (e xe*) Apgupiag eroperey om 4 qu (4 113 dee. Ae 230 LEO (4,0) fu o) = vq Un ¿20 Y) follan — Laida is Pe grartak incrunaing Asreepon ar (0,0) Using Ae progecty nm. 2 = (e 00 )= (4,03. DEF + vz lo,0 VEA (4,2) = (6) - (A 2) we com we Messina ROBA Me Secona partio, derivajiucs! majo OR E hq=/24 AE exam y a + ES BA partos es PR or z 3 Y Bd 2 + ES OS Lo ux el Dx x »?3 > ? o dE 2 6xy Say Bxa xn m Dar oy Korice 1 24 23] - 3 21. 0 at +]. e ax ES E sy ES Day cl E ECONO Mic APPLICATIONS e TOR MOJO y Qi nd Ex: d >» demand Quncison ; d = A0D- p% 5%4p +45 . Mean voJe : 434 Ou , 0) (63) Yi An = 20m -p* e Mx = express He vale A Magovát y or cade pan 2 5 donmun Peon Veloe 08 y usd espect to he xi Mariola is He Álisiom A y onto xi. . Harpinol wal defines. as: LE qlo -Aei) > pue) - di ; A con te defined. as Mere 2 a frenos ro A dr OA A nana En te expe? dios - 22 e ElasHiaty: Les a mensura AL VANO Tano. Etostiata A z Be > JURA ACI) EA) ia o aa : 40 * ES e — A00- p? co -p* P Yi Alia) 000, 0) O ES in Mae Xi warNolde Keeping Coustear 0d Yue Oller vanalales . MARGINAL VALUE PLANT VALE Progress oy aspematos A. 4 ospermarialda ak 0 >a0) 's unique 2.4 4 Alperralds ar 0 > fis tonbavos CS Aa Ñ 3.04 asqprotalea aka > Le la) = 000 Yu h. 18 2: RRA, gis di gero able 34 dina TER: dee qe AR be conBnwovs funcion ak 0: 3 a and Hey are COnFwoUS de 0% Dos ig Vista." 24. > 8 Lis A are CONPTumus te E dass ig Wijrt?tio. " az ond Hey : Ls Ey Aru de 0% das ig Wijstt. 34 ona OS PROP: LEC OS Y digfearolas ok a, TTHEOREM OF SCHWAAQT . 2 oq ana, A _ 2% MAC" => W1t),) DDR DUDA OXjOXi igge c? > 4300) is apAoÑo PRO: y pe E o z o a q508) = 0% AL(0-5 = La NA E 124 47 Eehomowr SY 0 funcion ar Mo ) a gm 50 mM EN (o po, Y necrosis occ elesat or a La 0% «AL Z (01% Q 0m0 £0) 30, Y decrenes ÍA or 0 “4 qm 20 0d EN (920 , 2 AeLmOsts acc ler o a. | Ñ _ or A cL gam>o má $020, Y increases dec ea OPTMZAT ON NON - CONSTRAÍINED OPIMIZATION DEF: Formes a 208aL vabaciww 0 a if Nx iu Hue nexahle0or honda a ib: 3 Evian conos ] d + E $ qa) 2 qe o Ñ y] imegualíkes Ort grace, woximo Lminima] Are sino 6s Wwe, [gim € 300] So, ig Het existes Ye (o) 0má Palad > 25 (a) =0 Md ts (0) <0 > Then, iy Hee Exises a, sesave escrreas wee ata > Ugo =0 [> 0] DEF O: debo Hinxa ER Sat O h is posiive Ex 02 gin, 2 Yo,o Ho >0 his pose Aegink ¡8 Vo, oh Ho> 0 bois negarive dam de gia 5% Wo, O H is sega dept 14 Wo, vrrho Xocal UNAM VAYA ag Pa a Local votan Lon Hs yu > ag (0) indep >= soadte point Lpunro 02 ¡opoca) qa 005 AN ES pá nes» . GQ Ex: Classiga ana Ja codiasa quints: LC a nde xs a rr AX X IS as Y Bla Ax 20 a,= Lo,0,0) Hgo = /2L Td 2 Dx, coa (Q13,0,*13) >x*, 7" a Bl zamo so sos > Ll 24 Dx IN Bro DXz Midi Xi 5 X3 > ¿mn 2er > de 2 alarma o O NS Ce bx ka: Uzzx 3 go Lo DKSSk, BARDA DX 2, a a ES -25 [2]-0 2 0-2 Noy L osx sigu > Dx dra BR Lon *Woloxz0 Ok ba md Aa 2 3.2 z1 .24 [2] -0 >rr DX2B%s ki: LD -2 0 Cu 2 PS [21] =-2 DXxz DXJDA, dk =2 =2 +44 (0,0,0)= 2 0-2 Uma 44 Uta.0,92= E Los bh Ja tual 4 b Ada Lal =4 2.0 0 la] =-% 2.0 4 ma] = % ye . E om Ale point ASE focal O => mó q ES O aiavta INTEGAMNTION BA CHANGE OF VARIABLES 34 comsists or pi He inttgral osina, another vamo tu Ex : a dt = u' ax — (61) 3 Lamar to lara > to 4+2ax ; Et x ¿an a) We Sulosh tute : sy , ED e IA O TEE — OR tte de PA 2 za At A IS xa we nerd o La-woku the charme 3 A += x al (eo e (1+2x) Uma”) Tasax = TR y SA > Ss 2 
