Linear Algebra Summery, Summaries of Linear Algebra

Download Linear Algebra Summery and more Summaries Linear Algebra in PDF only on Docsity!

CamScanner

CamScanner

a OR i Ce al i eee bg ~ —— ae 2 a _ er —_ _ — ® axtrby=c, yt “we, + ne sol?! —vinconsisient,| » ncac) = e)c *Anvese . FANS xn matrix, follorng Statement asx +b wl oy : ACBtC)® AQEAC uniqueness of mverse matrix, Gre equivalent, one is tre 23°C. y - 23 ate as cconsicte at | nee ae iq yene = attri» hcg siier i al © unique % Ba! ex ws | | D Asisunvertivie te @ N6 seh @ ionortety STeaeegese . I+ wen is invertible , A exist, 5) Ax =o-has—onlystnvial=self! xe La pavallel => ORY - Ale A'tB -| (apc) = Zeta! €) Te reduced row ech, form of A om lines, Pa 2+ (KA)T>tpT ber’ = Whnke 2 lon b J : . » Leasey Tx . ‘ =1\h lew can be expressed ag pre uc of E OrndientsS are differedt. + grod are J d (AT) A (A) = > (A') ; A's La) 12 Ax=b consist fon overy ax] Sam - eal (AB) b= elaT oo AhtS. ar 5 ! a Fy ec. A n ATLA matnx b, a Pee y-Teterceets are dl. y intercept = avi br Pe a1 = a, are Same. *Gausse elm > obaat A = (ays 4) ax=& has exactly ene sot% ee ' ec op “ ‘ . a Medwees br Ba Or =a, «Gauss Tordao for a Tf nis invertible , -_ t zs ied on matnx b huniars cr . a ; : Be Ly reduced Vorper I, B' is invertible & Ga') sae “= (A) +0" i. led iy ba C1 =Cs row omtm | 2. AP is or & ONE Miners and cofaoicrs ; « diageral — e somo rm, bi GB, V1 Boon V leading! - column doe 4,kA is & wets 4 <]2 3] = 2-6-2 Hien” celumas [INT Om eb ahr I es: v 10 76% C Ket) 39 CF A) ae | | 5 ° La ; | we “t Py tomers Ho neais L.S ¥EWte asquare mat =F AN A= pn°nz e)" Cu C1 oo, \ T danonal [N] ——— det=f=d “un =| aK: Adj A =(cotacton eonilen) main dag e - System which af constant +e em = 8 ®* zero matrix - dragenal ‘i . * Symmetre = matex dy, = Oj; “23 ‘ea 5 8 et [35 [ss o ] A=A 5 oy “e skew symmetnic aj = -O,;) aAmainx is s-s if QT=-B “5 £ [SE] a e) | 7 * p-1Ca+0™) +L ca-at) 2 a ’ * A+B =O+A Com my , (A+B)+C = At@+e) Associ: “AaB a en C de net commutate in general ) “Azo pot Impifes Azo. * RB> Ae buat BFE a Properties of Scalan mult iplica ften j.kKA > Ak -“ABEO ii- FCA46)= KA+KB oe nok m. (Ktd)az EAtAA necessary IV. (ka) A= KAR) that zo or B=o. upper, bwer are Zero 0,7, +O,7%, +. . “=O, Ag™O,.. -home.sy 5 IS taw equsilert- to Idenligy mater Aya, oD se +[2 J [ augmented matnx ec home.sgs wrth mere unkoowns +Hhan infiodely many sa%s . eq”. has Mr=o IS a trival sol Céol* BD she2md og 1 ) -4anIn= O 6 Tver: Tt A isinv Prof : S.(KA)CEAD = KE! cand az se_ot Transpote ercdtble , 4nen pPTalso Mmverdible and (at)~!_ (yt i A* cat. (Ma) = “, zie Similarly CAT AT Ba © A nxn indi Is — _iff det M#o oie | S| Elementary Matrix Oo Oblained by performing do I. Arn 399-9 T Am is CunktEnewns om so ar na) Fr elinear system damm mere unt nanos Ce, EF, & E,) Az=T 4nan eg md e800 HON AL infinite a macy Heo , i * ar-l- léadtng vanables ae y> 2+ A=ca)! iol free vanables . 7 =2 f sus has a unique sol Agr an n= Sol vector , (ool li 0=), a centradiction no ap a incensistent ,_ : | € 2 + & ” GIT) 3-3-9 CTA‘) ‘| The nxn matex A is toverdible, ret A is rat equivalent to tdeotity GIT [ooo] y) A i rows of zens, single vow op, ” oy m. JO Teen «obey ~a)2 4] + Lt YI Gi Ca Se ded Nom ex PO. aS O Aj | raf 3 5 4\o det A= 0%) + Ata Crat din Bh 4 ee 3 nL 3 ses + ., @ det of upper BO, lewer BO matnx = Qu 4oo 993 Vay b Fees | o (0 z oes didigeml> da erp mode (6) 3K%3 matcix i [ae é + square matx ! 5 @ “ee det p~ det.n" Ds} abe a b d ed e@ ght 9 bh a aet4 bfq + Cdh ait ~9eC-hfao -idb @ Tf a Seuare mati has ‘raw iovertble of zeroes or column of gero, —_—_ then det ao . © camscanner i} (@ . 7 } s < 0M mw has eT Ais an mvetdible matrix | my ene non | lz 1 adj) ze em ry dein) eed: Ceg]" A Cad) N= adet HI (to prove 4 Cramece' rule ‘foe oxen matrices, Cendilicn 2 nx n madi wm for nxn mainees () Single mew Ss ont eclumn Sms *) ha) dd od dave’) | (34 aev, there le a vecion IFO 7s OAS Eh Core 4 i} Span ‘vector Space V are said to ———=_—=——eEeeelti( —_ Ved ste r Vare ney ( The Vector Vv, vy —_ Fel flu supe me _ 77 YM ita be Span y i ¢- ved ‘ | =m such 4ha + mteeryso |e ne” Pada Wwe tnve vse) 4) *YUEV, A) If a yev, «is salar ' A(mty) 2 oma dy #) (o4+p)x = ta + PR n4+ys Gta such that v Pas < linear com bination ef them Thad jc every Vv PRe=span L1yx,225 veoten in VY con be w — miten asa fy, FEV, there (Ss salars gy v= OM ta y 4-+. tCnV, H 10a ~-Cr | —« #SQan 3@ ov,et sets naved nel be a ftxed numben O= Ced09,0)> 1,x,2? > a) &A)x = (9) > for every wetor 2eV, in V, then 4 is called a_basi's for | AM STH C16 mu iplizatwe’) V prouded that edeotily a) Tne vector$ m $ are linearly K-Scalar—00.0d~mall iple-mg. 8d ; del A #0 5A BPG. 1Al= KIB] 1.1! tind det ® =D Ci) raws—«interéhange D0) , det. at 23 3] c . 4 oO % “gay 8-[2 5) stots | det Axx def B= =x aa es I, _ D, | WD Ase ew Dad ‘seelar dows 2 x0 wl 2 67 > AX» od” 1d dpxrdadd 245 Yn Po AL a8 a D, ! - matinx 6 & del.qyou.cdral.onrd, Tee 2 | y* <3 | Sn Sr [1s Ontky, aytke, aistko,, 7 { = - Da- |t 44 ‘ Z = Dg | Bee ial ee? ——" PepectiecS: TF AR is nxn matnx « det CABD= deter). det ca) e det ck AD= Kk" detca): Vector space . i Tre veetors af orthegcoa! Cb) a’ Subspace fsa bset W of a Veclon spare is called Subspare of Vo it W itself a vecton Spare unden 4theaddilion & sealan mu li plitatien detined on. WH4 AF and wis ao subspace ft 'Duvew DP uv+vew 5) K IS salar, VEW— EUE WwW "every Cubspace of a vecter sface 4| cenfaine O, U.¥=o v edad CATH Fdet+ Madeteg |} vi aly tw ts 0 y | é Sd die lb a2 = Tele — ; wes bi Ss invecdeble , TmeN det A det ya. u.v = vu com 4V Aull) cos 4 | | 3. Uso tf wsro Atop! A 1s orthegoral => [Al=41 m " Vecton T vl but =[AI= + ' Yama | A Is ortbag ona | ft az0 h. Tay G 3) cooverse iS oct tre. [email protected]. me Cs.t)¥) - 7 1 : . « T¢ and B are sieprlaremadniccs 4 Orthogeoalty of raw vector and sol fa fed Ve Maan > Als insecaeble | His net O subspace, cuz matnx is not there , © v= cLo,i] f: pe Xn 2eTo Adel of real valued , C4S function defined on Indewal (ot) , cn dor some in ve rtible ) To prve V iS a vector space ‘) VEb SincedmO=S is de where ! m for” — Pp. Jd= veo Ccloseire unden eddie) } 1et a heV > Am pep , sy Biy nev and yEV , Aten mHyeV Oe ig ots fig PA- PP’s P A ; . Since. 4Um of function rs cts = 3) tate any 2EV, aisasalan aelR 5 ath eV PA ~ BP ox EV Celoseire under Scalan es Ith) SoM ic o.veetp ra pace» det (en) = det c6P) det @.ded Dua det 6. ded P det A = dele of . “B) (£40 cw) Properties + my 2 EV (Catup2 AA CYAD of vector additon, «) for alt agsscciaiive law ¥) There is avecton O€V Suoh that for all ne-V inéac cem bmatien Wey tGavyst-s° W isalnean wmb nalien ot Vy My ie Bat Vn . +Cr Vn . Bases don vee. SP | 94 V is any Veolcn spare and , S= ivp¥e,-—-3Mn 3 (8 a finite Set of vectcrs b) The vectors in 9 span V Yo asedot & vecters let fa $a men) inde pendent and it epreude Anat ) Cyv, +, Wed -.: env, = O has enly +naual Sele CScalars Dsomd o) IR? is Mean lincary dependent, u Any hnearty independert hb, vectors tn 12% 16 a basis for IR Sn fewl FO. —Plinzaryenindcpended, OPC ave vectors. 31 b, Cy Ca, a, 4) ds, Vand wine Vecterepacts over HM Tiuyaw a tuncten fom Vp one | e Vtow , Tis cahed linear trans. fem V4o wo dmet Sahstics v TC uty) = Th + TY) ¥Tlau) = 4) a ee 2 eee linear Trans-formaticn . Ca dditier) Csealay mu tty) ( bef] CamScanner