FALLSEM2024-25 IMAT201L TH VL2024250102527 Reference Material I PDF

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This document appears to be lecture notes or similar study material for an undergraduate mathematics course, likely focusing on inner product spaces, linear transformations and related mathematical concepts. The document contains various formulas and concepts but does not seem to be structured as a past paper.

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.px - 5- L.,)"-fo^{a^ la,r t@q, -^ -6 i:y-yi:v: r |- r.;::-;). i-) ;"V t"-J^'t- 4^ !., ;(}- q_=Cj.,rrri) l. T''"* 4y 4 TL- i^L y tv,,v,;Y =...

.px - 5- L.,)"-fo^{a^ la,r t@q, -^ -6 i:y-yi:v: r |- r.;::-;). i-) ;"V t"-J^'t- 4^ !., ;(}- q_=Cj.,rrri) l. T''"* 4y 4 TL- i^L y tv,,v,;Y = f"'-J- 11- 13/ 7:3:, r f- - Y =x-r 3r -Yn,Yrt:' T rT n- -ohtir ;,*']"^tt- + lT X' A (_ c\/a' Z A-,U ts " 6ku\e-/ :r, LJ, il,L cr,oY' ffiq ?. o Dl*'':gt ? L [... A -0 h*o h*^nLL t^*" wJr.^r,l ,r- "^A U in \€r\o^P't- a tL- tb, + )$..*}r r:^^"i'1x,3)= "-) i; -Ytl =)lv -Lll ' rr t a ^:^^ h"..reLhr'AI' e ^,) v lnK 6;,r + )1" 3 = l[ x]f llYll cs's 0' 1-*" lru.h9I\ ry ,il,qo^o[ Co- il ryd ilr*$-ro ^ ^"J,. I Fe- v\ EZ-*rf"UJ ,, i,) A*Jrz7 = Znrzz + zerzT , CoLhhr.lf ) 1t:rry 7 = l{ ZarS 7, ?ff fremof;e,na)ry1 ?v) 4x,tL>Zo ry.\ nJt lt )L:o- fl Pd" (V,, z- ,-) + a C*-[) \,.eh.r rtpw-e- V a^ol 4.^ Innov P.*J^,.cF Z'/ L cJfuJ. x@) ,SPc-Le - i^ |n*.v h"oh*['ELrLliJ"^-^ pxhc,tr"^r, tt- pe, CR),) f^ c"0hl d. n-;PoLe_- 4* W b &bljr }) zg()L> = Z-),.7o > r&, L", J- z-YzP 7 a.\ = Zl. r\rZ + Z-arYt 7 !i) z),t ttY v ="' lpUt pL,7 >)t_ /ytsL7 =, 11_L:f r g Z. _e-. zfq,*Eu,v 1 6q -7v-7 'i' j;" "'- ,/ctr,. r Lvt Zev't Lrn T''+'z -7, ?Q" +v>7 ) Zet,iSw>r 6V -Vw7 = 7+ LSVLr-'7V7 +.Zftu>,'6Y> Z-tl/,va 71-+g zqrrg 7-Jbzty' w Zwr rvtz -.3q =3o F ^L 3) Exp^i[, l/&* -3 rrll "l,^]-f Ara-3 \r 7 ffin-:o,tlt = Z. :./^,- r-3v>=4- Z-\v-l'4^2,+ Lw-3\rr 1-)'t Z- l9*3V |ff"2" = z*w *? ryZL*i -'3v>=+ = Zwlv= ;. b Z\F,n,=,nq''z*ynli-.r. i- - - 4 Zvrw;2.--L rl*\*,g. ,t o *., i"*1.1\r, t7 ztn7v7 -lt Zltw = ol '.. ' = +-1 ll rrll -'.= 11 ll*ttL - $ Zv,v 7 i. ,.;$,r, rl,off: ,-. -.- >> )lxl)L= Z+ra7 'i' -- ;cll =u'','r'2._, ={'z't'tt7 Il;clf Jl - -r /hty.zt+Y r)(+Y7' ^t.,.,nL=.21+yr)t+yz, rf* = llr+gll =lzx+v1)k31 -"/ " -,. rFy{ 'WX'H#, D Svr :t;{-"' !uz*+e-ur-z:'^-+\- :) Il+^-tv-ll- r Dll3u.-+AolL ) ll )nranr "P'-I*l^ rn P| Vdtl W Ii*- !'ll'^;^2 1,,, ^n {'y'' oL.'"- z)'t fl 27 ta 7 'Z^, \r ;= * r !, -{' Y' -avY ti3 = = ' I'.,t \5 = /-j,Y L7' I Jfi snl- L.t- [^-$-= Z Z, /ZL7 1T-r ' = AZxt t^L=r[rh, 1 4\'ttld = Zaxy fxyT+42, t bz&7 = Axttb=, 1 alytLzpT* 1I'" t Yz> Z c'l;!bzr) , CY, 7 i) Zoqab"'r-, G^r +\2, , 'Y) - lq +L=)3; Loar+u'illt =&x,+b=)y, +-?[q U]a-)YP' = a,CX-.Jt -Lt |r-LrJ,+32LLY) = 6- Zu,Vz*yL Z-'*xrw 7 '.- 1 t''/ Zrrlw zrY,xt --vrlt *-Ytx,L+3YrTt = JrY>-tLrY,t31'YL )ta Yt = Z^,V7- ;i , zrr.,-r \r7= = (.a)r, -Gx) lL-turiy t+3Lv1')YL i Lx, Y,)tr ,-13*l/ +3{'Y') I I I =lt Y = lt Zqr\r7' i I I it/i Z*r v Z - iL, Xl- x.1 il,y -rtr{1 "\- 3l'LlL t- ={, LLr 'tt-t#+Lt = Lnr-*))'+Lx; >>o. c lA L'{J) l-le^ ru/ L*1 w 7,s:-r Yr '4r 3;\vltt3xfi> Inner* r\*.f- 6l'*c€-- P-,* q V.^.,PX W Zrn,\r7^ 'tr Y,a>):- tA."F^n fn Lt= /-),t,Ll->/ F*A^J- ,spote sA P-, *[,*- = ZVr tY>7- =:B 4l 4vv- '-Lrh Y> L I trlL\ 3), , $= Cl'))'. "^A Zu.r\r'> = ZCI'il ' U,t) 7 llor',/ Kw = Lg,L) ^^A lt I r Lt,t) 7 ,/ lt\x, \l- 7 = :- L*-b*l {t:lz -1 ' z (*rb) =l*3r =3 -o o - "& f-^r"^ o ( kc^t v) = Lr-3 =6 f,y'.n @ ^A 6,. -u. ,LPr,"'t- fb t'th aA iono-nr P\n'*: ,, ) Zwr\rz rL F c.n iottlrf ta'Y--a33: -ft, n*k Zw't: U7 ^o =tt ,^.d ) u\= z^r Pl" xl7 t ffi+--'u^123, yrr\s7 '^ll'*-.c- \r= ZI , - / tn' v 7 = 'f ik x:q --9x I = 9;[u-zr ->c._>€ -o'.o- uL \rn.:^, Yl 4*,. * A1 gl rr..-t VZ t-i notz;'\7""I 'r! \ V-i*"y & -"r :a*1Y' I L Lt\' + Ca'J + L1'9' t,a l-f"w, sL.u fi^,t Z-xryz ' tP 04 p\ *yq- \= z)tt An \rnnz-x p*J-& Jp'-ue ')127 n^I !=Zt,,Yr7' ) L* Lt= Ct t2-t+) , V= C&z *Zr;) ^"1 'o=f'r 'L'4' = -)F" u) [{*l l-: z^/k 7 =l\ri+L: e-1, vi) ll ,-trr fiL = Lwrr l r,^'+\r- 7 = Z-L3r-\rq)r (2,-lr9) 7 = 4+t+8 | =q) 11 *1rrl | --'.Jql - r) ry L* {,,3,h Lr- ?",-rh*, in v- i) z *,5 z ='\-o P.43ot) dq =Jo 3bL) f@ lrt =23,&=' r "fi) z{+ 3, h > = Je LO rJLr)LLn "ol k dr = { *@ hLl)ar, lt LDLtb) ,5 O =t"ZP;h>+ 24,\=* ii, z,.lrg>= t'., [0.?1il sLr)rt ol =ft lfi =* to {.. =[tH,LLil dt = @at1t 70' iu/ z+,+7 td I*" 9 Z{,g 7 l} c-n inr,c-"( ta It ,E.13 ;$ ** **p, ry 6"llr"19 *, )n"- h'J."'l- ep+e- "' nn[-( I }'trU ta'Yz.- e i)) Zs.,\z z ;Lr j../ -- :, ,, ar \!L_rt&y ;i) Z-\,ry z > +)ttY t+\lrYs r iif) z>lt\ 7 = x' tYY-{'Y t irr) z-\,r3 z =)Lt 5tljrt2Yl- +) lLY z V) Zst r3 z ,.-dr Yr -'LrY'-lYYi ,\ora ,:.",::^ P"ue h"r, Z->L.\=-L >9 x/ zYrYT t ,,\ gpo,*z V, "'' G*l r n A t ZxrYzl *ll'Llt llYll' ?*41 Zx'x7 L.h -t=or il*" Z>tr5 =-L '! ().- L.€- -/ Z_O7 O:7. z-wJ / Z-o-lo=7'Zi'Y'7 -\ ZorYT ) -TY o4 o'Z\rz=--l', lt-o" i: t-t"e' )L=o, lLo,. il- h* x*''fr- a'nZ A"'A- f' L",\- r^r *-"- we-kanq o 1 k-:r-tgz bri+1 >' : Ztxllxz+ZttrlT 4 LYtbtz + Z-vA7 )L z +-tzc''Y 7 t tsl'v Y 7 + ZY'g > =YLz^, zx,x-7 'YLlZal'tz +/-vtg >- 'illa^ =f r*$iu hol M ".U*4 A,x>b-+*)rr)V* , Z.Y,Yz i2 F fult^"^'rJ hot' glI*/ no,.'*I';'u,' G^4 ; -''*^t *r"FJ r'on I+l-' ronh' v*'P4"9 iv': " -faiihva' i,'tra"""f Z>t l y 7'- Zlr)L;> z-YrY 7{o l,+prt e/ w*nl. J*^ ) B^o\ ff" ^nXk, Zxry7=2^tYrt3lYY, = Cr re) A !f =t\,o)- s;!t*' = C");0 = 4't:? 1yxfi ttsil Zrtr37 =9* xl x l+3 rAxo = A+o sl- flrrr -@z ) ttrrr :i,:'r:;,yL G Z+l 2-=)L-t ' 7 )lrtll -=ffi' L t\r'7 ilyr )'W)= Ly,Y7 = L*txl+3 *o = ?-+ o =9* ) tl Yll = rf*'- tn' =of l cla(- i -k.) CpS 0 dtLr tlTYW , l _\f kl (-e,s'(G). S= W rlt +s lrrkurn (l r' n.^[ [)- d'lsq. [.n-= e) \-=LL, lt;) in rr_ $ry,ZW\r >, =*-3 =J +lo a> 1lt^lI = ll*tr2- =Z,L1vLZ :l "(+9 =lQ' tLgf3" \llrtlt :"8 \ hfl, -: /v,v== k,{-l r) n n./.^,1r7 q :. J' -=.CrA$:M7+:t' =ffio tF tt *ll 11rl I \-w' c:'2+ 'n h !'W b-y fr,t v*w 4V*ffi;ffi,,b,n ry ? , r I^ *l fin/u: n i) h=cq/Ttqrlr3)0'^'A ;,; ; = ?;, ,i *'\ ": f'' *i ':n P*' 1 e 9^ ti oA l t o.A r\i) w= c3'n'n,,":: tr = &t' 'i "--;* ' n[ wfr i^ ^A 1(2 e't' ;:tr ffif,#)I?.r"frYo5 3',Y *I nJz \r *=c ,3 ifil ili) ll,.rclt = )r'I Ilxll ' iv) ttr*y'\\ 4-11 r)\ 1\\ 711. /'*o, [}*^ t,,llll 70''^{ ["nLe' ryi+ ':-ffi 70 -> l)c\zo = z/-o p7 = )rrri o li) TP rL=o t'/ , ll*n Zx'x7 =f---- o = o- +' lt:t-ll=@'x7 ,. --i,*,, :o -r,.lr ^r, "'f ',t- - A A nJe i4 \4 1=o ' ',i, gnLer l)r.Jl ;tr:::r, ) llt t)l i lr'l 11,'ll' iv) ufly,',i'^:Ii '--';:;:;- i^*2*,[h' w€-.bkil'" a irrlH' = z1t3r )Lt3 7 ---Z^rx>r Zxt yzrZii.zY z\r37 =Z)Vx7?LZxr37 +Zyr\7 L[l *l )'* yll xilllstl r]l y ry r- =Qxtf rttyr * ) ll xrbtl * I):tll +)l5l)- b r) t-* k= LE,s,Br tz! F,^A , )) ACh,v),, g\*@' -- &wv)=I*-vll : J Ur ,\r) )--) = A,T\t ) )LL -Y) l\)r-y ll = -xll - IJZF-)L+ LY, -xil ) A II g@ = =lw-x'l\ ' :)rLtr,5)=l1r-ll\=\13'-r\\=J[YYx) --) eL Q,x,l): d t 9' x)' \^k- 5* 19 uen"g ^ y"i^n7*k!, L'.,d * 3 { fi,e *dvt')+AL.d'. F.,",".l^oF 5 1 !_ v :d 1t *g Lt, z'Y,17=.jP.> 3L{-) yr*',:fi, ;g s lUn, rt ;i)%r)=rLr,3u>x'l F' ^r\ A. ^ilr' t 3 Cr.l >)- q tI x n i (6) ia s.5 E ;*-:--:- J.Uoryt r 2ft,)= =j p)lcr) A" o C"J b= z-w,yrn7 @n1s,;tt i) zwn,?c$ 7=J Lrlogu)M I fl-r Y WAil rdT: l' fiIcl) h'=z3c4,3ci) 7= o-{ I ' :- !- 'I \ es A=.h =-4 - =9_ rW o) cx./ YE A= c.,{'(8" [*u* *"r{,FOt n ii) Wq=f a^A !d=)t rylrr4 a*€- 4gv{ /_.futA 16 kY V b, 6-n Y)- wtl-tY Tfa^, odl*g*A h in V- T\"h pL, ,=o Vy in Y f,]o , Ar-^V, Z;{rY =so $--r JX , y in V-. 4,PnrrE'^'r L M, r-"-t- Zx,!z=o {'** ^!l Y inv" 4rA7=o :> *=": Tx*^,'.Hff,ilf v. Tl"^.-r".ht t r**\ t* { Ln^o F: ^ qin /=h/\D,'r^ - oror6 'to$A w'ht I inv y'ill fA-t=-io' c, \tr "^d nJ) t I )_,e , ' L)tr \LZ =o : i z--1 / \fn )z = o - lln't ' A a-tY7 n = l>l Z:' Z^,vi7 =CI P'* an) v= o t 2l i!;s 6v " -L=@ '( -) f6n- @nr*^,'Jl ,1 y )L=o => z>t-,fiz =Dr +iP) '-^ h uryft.ilo ,L in v o*l]''o)o^A I ^'*.h",)n d - br-,rnx *'htt ui r - t+e'l&q / P-^o\n4-^d w L'L"d c*nd, Y rt^r1lqtt ee{A *il1':l,,TXA ?wb }/l* }/l-* *l*A - *f*hil-I Tk^, z,r,y ==il r{r,,, t_q rtr{r il.rrt r,Ad, $=,, ,:) C,e"9S ': C^e^s D- = o, :> tltrYllL= il:rtl'+ nin' t' ^') J c.\nc "XII^'5"*A wch*AJ Z= TL ,*aNSu\ non -@*o u.#enA in o\^ io L^,,L>r V ar€- ,#*JtT h*/^.II- ;\f,'Le LW.tb *['r'fr e"A *,I-n" I). ['*^= W^'=L' \ t 4* tll f' b b1 \; --.0yil."5r"4 rqrhri,,.= loi^ t r r;l : :r,r [a, \ Lor, 43, q \h yr) h PJ TL *a,hir @n, + fr 4il"'6n 4'",,"r( Soil^ WfuV o^ a wrh ''r ffatu V ia inw"t"i h il* *'h U DF ;a.**/ r^lo+nol"'b' xl"; I}*h \^LJa"A c"l "'mn +) fr A\{d=h,vLr '{'l'Y^-ti k* ; ,''f1 ^,wo,bdt \naclu,r 4 6'n 6.n inna-,r t*-&*F frrure-v. ,i, [;[*^ + r\ Ih ff=G,il = lz'*,t b3 t f".j= n. tp^$Arx, /h. h; WAw,"'e +ry bn*n wrbxN + A- fu" Lt,Lr-, r r Ln &P''|- t O F-r+Gqn'i =LtQr+CyLgf l! ,i3 ri I B Lnt, c= f ,t civT tV, so k* fc]" =lt,,Lr,.-,i L'-'"r o ho,.no1'gt"-o"} ;yl"t1.* Ir"r, f)ro'"q*J'i"t t: j^'*no-'n'! o = ,c1u +ry !"--' ?*"' oil= tql oCq;", o =4,,cr t' -' + Q,'L.=-S ^ - -'t4sn'n=F,o Wrj =fdilq,L= ztrt'L 6 =QsyLt ^ zr' *) r''' --f'o ni'i =fr),*'.n:= o'1,o n, ct *QnnL^ l/I^i L't V h n \'re Yn' a'[s'*i':''u;l ib' o uL*, f*7r=Q;' inrv:rh.L.k ;:' ;i I" x^ily g*^"r H;i tt( Tf' J;" b"'i",s : :fi Tfl f =?: " oi YL- col.^_n \ { 'fr ^ colunan'L ^n_ :)/ a,*i=f ci',i r=or rr,-.a h*".^lal.T -'n kn..9.b*,'.lX i^lu7"A'*'' T' nn, gr 2h I '.,r, c"llt ,l \^ru4r \re.] *- f$r* A , i *^rryh f\ l^ j=it4 ^H*, x' b, ,^ )-^tt \ ry4r#,n' ruo"-J' ,. + ,. wJh = hr*' :a q n- p\L*i1 wch"'r i"' n"_ 'Sy'rc- B" D **&r-;\ a")' w > Lr, I r -t) J n, (\- )tr,-lr) I 5 t^ -= tlwll Ilul/ ,ffi 4+l tl =\)l- ! llvrt=ffi.i r^' 2W W* OYn^^o@ A --./ \to ;^ ^n fr tr4 fn^orr \-- *rF* d"\z' rvvvr'l-' 4** v fu JdJ"fu-'; x-J-rt rrlrn'"-J,f -: ZXf ,Xj ;=[] v t 'J \'' fih",ilW ,=[?','!nf+', sL t+ ;* t^,,2ts- ' r *T' -,\,\ + il- [':^ bt*ta ^A cAIrA t \ an [*ro [r" V if ^ "rt]^o^or-A a orht^*-A ) $.q, ryn* P\br^? ,bk S 4 *'*: )n BY &^ \- *1"' 'L ,vr3 ll L=l+lt:'-n4=87 L= *t'7 l'q.r'*or: 1|ury)l :':: ^;Y-f Qr-h) , tV W (^n =U,lrA J*-ch \ _a Z =O_ { L^rr)rs &, s. \ )ro) (tr)r) \= Ct, t \fL = L + z=s, \ &t}r b=l + /=) , Z=O w Z =-l =) "1- >t / n.f-l L*,: y^ :- -,tt z =o, S* dz-=C Ir-lt o) --, wl ,dz- fu*-; ^' 4)l -(k." W (;irn fcr;a -An0) Ir o*f)t5 o^A- I co.E b) Ssl|fY s"), : urer[ -_ /,SWcsbT = o_. (P'ss -- ---- ' = 5r'^& Ces'sD /-SM-)GinorFA0) *6r;, A-6o ilroAs--t'qall2- Y' Ar1 n_ )rp-i^ ^t*- I u [)", )rp-,'a $o fb ti , 6>a- ^.h 'ttrYi ,^ ^i 15*.*oh}b l*oh}b - _ ; ;^"n 4*Ifn"6 = I - So, il* Pwa : T[*, |,:nna't"ir( iA o{n-$onJ' I i il iAn;i W fr tl odxl*'Jo"J?.i.^U -'?r , \ ^ \, -u fo, A* =-6 L +o, -lL -l{s G,4a-fk,E) * = * flub h o'Ih", t^ "$ r^ ovrx *ul^o' v' oiu ,^!)' p-*i * :j* ;1'*" u z\'vL7['t ^ - + Zt^'W>$ - i" - t{-, )L= L\ll,'7S to ^n *n:::1" W9; f",ui'i=^' -. ,q] -"-r)l'vn a\ i:"1"1 a' 9'*" ;tr ,il,',,' *^r*+, '*'f'"^' n i=l'Lt*-'o; c,J;"#* 4 S*o "'*ille.^"1. -r Z>tnw" rV} 7 2 z)L1v1rviz * an'z$n r"si 7 = o+ - t0't-:([4\ri'\"7*:*-'-'to " f". L:' - zrry nz ? =tJ - x-i. $nce, fu,$2, - - , \D,1 iA oxDo^ot".ol'- - -= ,^ z-rlt"V) -7 =- g' -, > =-' => Z't-, tfr ='Zwv'vi7 l+en L,at p*d- ffi, rtor" H."ffi'.,q[;+-- t "\,Lry'* u A'"8 W ltr) u l'.r a=u'ar^r- A U@w.. + V=U@W ^A = i' c\ l*-iechoo ' rA G^*r-* bY^"^4..cnn*i.sn T-: 1{-y {h';- r LA r [i,., r0 s*,LffaLs' ,) i+ ,..'A o"[5.)_=a- f-.-9. tf 7eSinrc, Yt* L bn/ k" T'T'T {* ony ^LLPv-*Y2 T: lL* & Sr*ficiqnLtt,trf -V rt=I" P*jochon I^fe \r54n't- b fho * =Tn,tT)@ tuxLr), o^4 TC.. tu") '--- r{ ?nu L,L+t.! eI-tI)@ &*,rtt-)" bf b W*, E-tD nWtT)=Io] a^d V=I-1, tk"rLT) itr q tr-t, nYa,(U), Yl^- Yl,* i^ , Ai s,.,,ui, fk/L lbq:ut '^^d rt*):o' B-h, \A =T-C>L) trzt'u) =T (Itr1) =T['^) =c - I- (,, olLa)( tfl'J ,\ - l*".-rly, if q Lt*n A*^ Tt*>=1" y, V :,h *' p-CC ilrnnb)) =b - (.tr- q + &,-Cpn"LO)- Jr. + V ;a )Ao*o.r fi';c h T- LiYvwCO- + V =T-Li1')texLr-) {u, ;***l=ttq=\-t {***1.,t+d ehfi@trytLil' tr}€n.-g P*'l' "f&r4 cJ'-fi , J WY:'t Im -,- €. lA h^^ en "rlhr.,"*""'t o- bt;,s ff"r o'o $fiae Y. lAk r[,= ?'-, fl\)t \,t!:{A -- z-'4 '''zV' llx7o- zaL, I z\ vt) bt Xt- zrbfi>S fo fx,,7v\ ,a) ,rr--rla i"lr7"n dl*f ' j :i ii Lnt-hnv^L fL- *'6n- ";4 I :..{ '-€ 1B i$ *.,* sE * ri \" llxr.- aL,'1>s F - /Apr$.-,7 W-,il = s $ * e r''b€- ,sr/\*^oor,^.ofl. infl-- Sln La, s^,q.1 x}lornot J y) a-o €nt ) *r:il ^A "vxl.ono*o!. 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