Course: Linear Algebra, Summaries of Mathematics

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ev cayey wwe nnn yep pe 4, Similare_Madrcices : sativewiers ‘fame | @® Canonical forrms of matrices | © symmetric, -Orgthogonal — and Hernifian maces _ Lineart Functional and Dual Spaces. | @pbiveare, Transforemation and, theirc Properties, | @ Matrix representation of lineare tnansferomation. @ Change of ‘base's, | ®), Linear: Roctonal and dual Space, dual” basis, second ‘al ‘he Space, ' i) An hilitorcs , @) Transpose of a lineare dransforcemation 3.6 Orethogonality 4 th 2aoirbM _iysbserro @ Tnnere’ pridducts ‘Length and oiethegenality ® Projections and Least squaness;!,..)., ) The Greameschmidt POA, snlgonis dl On Wecnsit efcrnpyaF ¥is a Orethogonal r sets, Pp or tay Hagen iv Y) Toner. product spaces, Soitoitete ban W) Linear fonctions and adyoints, 1) Positive opercdttoress) unitary) operators » 4 and nornmel operators ‘eoonmdnd xnolimiz ot @) The Spectral HRedrel ecinot Invinoand @ Aeelication to ‘linear ‘mddéls gn Ruri approxi maton . oA [anahomA” 1 oes ct lin€ares Goce ari HeaeHan forehas( worn FE yi, IQS xlatkp t fi (oO) Marti Sonne, el sell ey canonical Forms J | ® Rewelion Yorwns definite! aid Semi "deinite. forms - hiriterl\ ¢ @ Frincipal minors and factonade | rms atl Js RS oy iW sty of | 5. Symmetric Mathuces and Quadratic Forums : _ O) Diagohalizahén! of symmetric mataices, (i) Quadrahe foHMS, Meno) ban finch (0) The singular: valuendecempasition ies.) (v) Applicatons to image PROCeSSINGs dis, and statishes @so.0q2 + ubona staat: Similare Matrices od viteo ‘ boy ‘ bitn peng spl)» paul ih *ht has to Be a aha matrix of same ord eSaaationtbob vibe 99 jigk ees , Definitions Le+ Aand B be nxn sq mateloes , Kin) and +he matrix A is ahevihare de B # therce . exists an Inverctible matrix P such thats F'ap=8, Mathematically, if A is similan fo B then we wre, A=6'gA So, the real meaning of P'ap its that this is the matrix of the Same transformation ing different basis Which cons: sts of the columns of P and standand basis of A. Similarity Invanient: A Property of squane matzices is said +0 be a Similarity invarziafi's @n invariant under similarity ¥ +he Propercty is hold by any -+two similar matrices. Geni larcity Tavartiant spp eR ASS: Po er eee Si mi @delerminaht: woh and © ies) have’ the Same determinant. ote ml} dove gq es er iP Ay . @ Inverctibi lity: Ais inverchible i Pap ts: Inverted = AA< : PAP. Have the" Rane rank. G= *hAM v= @ Rank : » Aand [rs | 2 (@ nullity : A and pap ih jhe, same nullity, " - vol cnie at 8 asd © ak yi! eet c i A +t (9) race. A.and,, PAR, hove, the. same, ‘cane, . pup. In é&F +O is) cus “wateteab bra A 9 (6) charcacternishe(, Polyndmial. +> A:and:& ‘AP, have the came chanccterdstic isolynpmial..) Q Eigenvalue and Elgenspace A and PAP have +he Same eigenvalue and elgenspace. (8) genspace Dimension: If 4 is an eigenvalue of pap then the eigenspace of A 4o A and the eigenspace 6 same Aand connesponding P"AP connespording to A have /-the dimension . Scanned with yw Let A and 8B ane two similax matrices: then prove fats EAL AKT and 9° BH@+B4T ame: Similan. 1 uae vl pores Hloedus rout iif f 1» Solution: We knows - fe Aand Bane similar matrices’ then, 8 = PAP" me iL J | . BY BLB+I = 6.B.8+6.6+6+F . Of 4G-nW eh oe = PAP”. Pap” pAP'+ (PAB: paps RAP’ 41 = PAP sp AP's PAPEL = OP's Apes AP Oe eT = A A+ASI js AarEb aot 2 i So. AX AAS. and B4+B74 B+ L- ane, similast ,. bagi ae “(proved 4 Prove that, the similanity relatién'is an equivalence reelochon. ; = ANS Solution: To prove that Similarly! ‘abiiion ig an equivalence relaton we have 45 pie thes O Ais similar to A any @ If A is similar to B cthep B is.Similan.-to A ZEA ts simian to Band “Bis similax + C then Cis simian AOA | Lelt - me = eit - Showing '@ => ANAY wit gyn O bap A tJ TF Ais © gtenilare to !anothem matrix +hen there, exists an inverchble matrix 0. ai! Lets P=, Jnxn orkaet Filion! wp d bon A +L > p's ae T +§+49+ 8.6.9 = 1+0/6+S -. Nows A= Pp AP NS pe Oy ) Sear. qAq.4Aq ARS =) ot). a SA TI + 4a pr GAM [ » ag \ oy he} “hy » CEA “ AWA FF AnB then BvA => FEANS 'Simildte 48 'B then’ Pars triverchble’\ Matrix, p (Such +ha+, , ARFBP Cb belt > PA= @P Bn | PAS" = BRET a t 4 Fat" i apt SP wed aw icitalon +, AxB \ ot vial Similardys:i Q=aRAP Soh ow liais oy sulin SP? OSBAPE ot stolimis 23 9 arty =? PRPS APP hnlinin ef 5 aadt => RBP'- A » BNA > ES yi JPs i Considerc Q: Lek Feberthe linear operator: om Rov cud defined by ii Athy fer | tii ode fii TG» a) (4ois~*ea 5 dni+%2) MN ¢ PEitoc f S jodwity ae J ; \ ' wil int coagik f Oe e ‘ | ae a | ai ' bases E={(0)»,Dfand ~epnte douse S={(-3 >» (5) for fR*, Prove thet the ‘madre mepreseniatons, A= [T]e; and, Cuppy B= [T], ane similan. Hence versify, that p. @ trace (A) = trace (8) @) Aand B have same eigenvalues? > > rinih ae t ‘ - - ff fa. cl f < Solution: O= po Gian Hoek, AN Gs Thi» Xe) = (Are Xa 22.4%)’ ’ ot E =f(.0)s(O.If vd bs) S= Ie 3),(2,5)§ t ify Te)= Thyor= Gad= *Cua)+ yOny » = dy roe mins oy \ 162 T@:) = T(o.')= *(16)+Y (0,1) AR 319 AOD { HSy OS &) \~aty ac(th= [49 [27 heals GAT .ash T= T= XOI*+ GI) FE. 25 Lyq(3d= (5) = Caep% BAe By) o| |, | TS) = TQ. 5)= (3,2) ue % (vee: Ys (2B).. + = (meet +Ote Sy) J Heres | 2 5 || ro} -) ne2y = 1 Same {ED le £€ Iip- of 3x4 5¥= 5 ——@ " . | FY es = | +a by solving @ and@®) we ge* ee ¥3-8 | os --08 a Agains “+2y= 3 — @ lw - | 3xu+5y= 9 —@) | by solving @ and @ we get x= he y= o . = (Tes, fe a rae ak Nows Let+ P’ be. cas; joverchible male te e. sige this gives ; scrip | . 2 \ . ~ rs = ify co | 7 Fe ti S| Q 3 #5 | aan rr o | “| g -2 7 . iliac [Bs ab # i" | ARs OD -—5 2| i 3-4 gine [& 2) -(Ohisb 7 ij a det(a>wder(@den aw ynsald Orava pay organ! she U Ha en! tredice (B)#I'S40) = bk bloik sete odt atgve yay ca ry hy JINS | h yl ly i noryl 20) ) 2 fegeetey'” insan ) CiwP my tne Hh +t prgqney perl soliot \ . ie enorrongo ‘D) For eigenvalue % ‘ [xt~al's 310 fy ane 4 agp ibbp soto: v 0 (vj HUST = (v3 an ~: A- 1 x ooplpoe Kaw oat Laoirnandqitlyuc sol nae ou = (a-4)(a-1) +2 J2u sobs = aisare CLIX :.[aT-Al = > AL 5A+6=0 BQOAR BAZ yp il en esoinkelM a5 -3 |AI-61% | (e ty il sop? nd lessor ybyp od AA lS! R=! yal ye i Pay a! a Poic qn DN es Mare S ot os APNE s =, NE; B/AgSo aj cotitogv ack oy did) ‘3 re ail ct wt wode sW .Ceamulnd 2A » a. 5ALe =6 Peale itno lqittues yvinitpat “ 7 wes VAs WA + vi\ ‘Caray A = (ew) al '. the elgenvalyes.oF. A, and B, Ome SAN: verte] = ( 7 Scanned with = ; a : Linear Transformahon: A jkob » Definition: Lets U and.V be, two: vector SPE overt the same field F. A mapping) PUY is called a lineart teapsforemation. or, lineart Mapping) if i+ satisfies the the following “wo CondiHons. o : gulaviaspt ie me © Vector addition: For any vedtors sK PU They) = Yu Tey 2 { 4 pP-£ ©) Scalar multiplication :| For any scalar cs vector v CU £eri_g \C pg) T&v) = aT(v> 24+ KR K Matrices | Lineon ‘Mappings ; eS -ene SK Let “A be ay real Fixn ‘hate, Heres Ala-ral determines a mopelog Fa: Pai* by F(v)=Av Vihene the vectors in k* dHack™ ane written as columns). We show Fa is Wineare by. cel mahzix multiplication, ere Fm Cw) = A(v+n) = Av+Aws é Falv) + Fai), Fa (kv) = AKY)S RAV) SR RCVDE Pio Ab l ha tney | Tr others Worcds,' Using" Ato ‘mépregent the ait io we acti 1) apom: a rotheeasdio Thus. the matrix Mapping A is inet: J if aw f f Vector Space Tsomorphism: Two /vectore Spaces V and exists o bigective Gee ate and arto) Unear. ping °F: VU. The’ mapping F is then called an igme- isomorphism, between Vand. v. . anaiL a pet Lineare Opercotorarspet THUFI VEY be a Naearc treansforemochon, thea, uf i a | linear operator ~ U(P= “CB. A linear transformation ! ‘ Vil a AT Oaaoces 3 ‘ feaen: UF), to uF) 4 self be x rons ve) is is “called ¥ a. fineare operator. | ban | Le Image of tinea, MaRPings Let: oicbatebextbe a Vinearn, mapping ) Fore, instance, »\ UL is; called the domain of T , and vis called ‘the codomaia of T. A odt bilo FO = cuyr Tf vev and yeVv such that Tw=¥ then |v. is, called tha\imagerof vi under Tho ol ‘Mathematically image of Fie! defined PY. Im (T9='}-vev¥\g yew suchithet TU) abn UrdomolMarnt or f\ pac xiv kom ad AT j | | i P46 A0Msv\on! 1MAAGNHOMIOEL SDNGe MODS (- tH -JSV astNons c Mdgs1ond sf i { rmovo isa Cahua hanpy .- cf -ga9) Litt) actel mn 2konm, King, aed ot 3 palananteee ie 9S jcy-pnes piss va Pall] periqanen : Ly \ ran ID icy fii apg y oslo codonlaimmoc! one | Figure | Tail | SS Image, of, linear PQPPINgG,.; 99 Jans | Kereniel Of Mean mapping! MOD robo apo t bro hoanoer spc A SAV = py tt y : Nobo s15q4 et PYN be a linear. mapping, Then Hps'4 is a So hy Te ‘ tr (oy Set of all vectors vo’ in U ahey Satieh) cinoat Pp} lab BAY T(ud)= 0 IS called the Kerenel of ae ae 7 Spoil nlt | pllno - 1b 64 oe lat ‘ Ree (TD) =P Ur T(uy¥d f/ag0m Tins NiMH Ad At sll ivy , -_ " . ‘ mons °C Alan fndh doug ¥2 ! ¥Y hon Usy 40 Cyr Change of Basis _ , = mpmef] Definition; Let scheint a a Yasts of & vectore Space V, and jet gi =]Vi > Va ...sVah be another basigai(Fore eferences We will Call $ the ‘ola? basis and 'S' the ‘new? bert, Becavsel’9itg°& basis s'! ed&h' vettore “iA” ay. ‘new? basis in s' can be written uniquely as a Eee et bion” ‘of the. enue sits Ansgil . Pag rN FINS al J ad 1 (4) Ue rt ‘Le PY [3] tahemewoPy <0 thew dis Mam Calhed: +he thang siof: ibasigii-ftéomn the yi!) uc Old basis S$ to +he. “hen, basis | $s. lees Herce f is the #y27S pose . of tha -below machrix of coefRcients: as Me = Ayr +QygUat+... +QinUn Va =Qy VU, Og: Va +2... + Ogn Un Vn = Ani + Angad... + OnnUn rq: consider. the] fBlioming two, ieasen FR: S= fur sda} = sun>+ BSH and s *. P: vi Oe I » (vor 95 S71 x > (9) fd} & Se the change Of basis. matrix £ Firom ($40. 8! r ( #6) eX eter r b) Find the change, of bosis ‘Matrix @ from 3 40.5... a ye 2 OC 8= le Solution: a Cae) exe(eu)pr = Given that. S= We. 2Uaj = {C22» (3, 5) ss Pre Mah = }@-9 ) (1,-2.5 © express, Us U2 &S ViuVa “ Herne, | yh ey eas = 4 v1 = (2 ) e oy, + daa. | ath -H,-2%2 = 2 = jira 4 220-2 Eo , “yr see = hibms m= 22) 9 ig 2 “Aya 4 Va (3,5) = MV + KVa +H, = 93 ? mu (1,-) + %a( 19-2) ty ua lO = *_ > => %a= -3 1 = Wl = (my+ Xa et ay — 22) VW = 4(15-) = (15-2) Ua = 4 (e-D- 81-9)