lmcd Lnear ransformaon of a vecor he deas presened here are que general hey go beyond he radonal mar-vecor ype seen n lnear algebra Furhermore, hey do no deal wh bass and are equally vald for any se of bass Whle some people consder ransformaon from one dmenson o anoher (say, R m R n ) or from one ype of vecor o anoher ype (say, from a column of numbers o a row of numbers), or nclude compleumbers, we shall resrc ourselves o he same ype of vecors n he same n-dmensonal real space Insrucor: Nam Sun Wang ransformaon n LVS s a rule ha maps a vecor n he LVS o anoher vecor n he same LVS An analogy s an ordnary funcon ha akes n a scalar number and assgns anoher scalar number A ransformaon s smlar o a scalar funcon, ecep akes n a vecor and assgns anoher vecor In essence, a ransformaon s a vecor funcon of vecors I s also commonly referred o as a ensor or an operaor When we speak of ransformaon, we need no resrc ourselves o jus real Eucldean space nor a LVS of fne dmenson We can make up any rules o map a vecor o anoher vecor, bu some rules are more useful han ohers here are several generally acceped noaons; below are some eamples, and and y are vecors A y A y f( ) y L( ) y Noe ha he frs noaon above does no necessarly mean A s a mar (whch s rue only for he specal case when he vecor s a column of numbers or a row of numbers and he mar A mus be a square one; n he laer case or a row of numbers, wach ou how we defne he operaon o preserve a row of numbers) here s no requremen ha we mus use a dfferen fon o denoe ransformaon o reduce confuson Furhermore, a ransformaon has nohng o do wh bass Wh he A=y noaon, we do no mply he ransformaon A s a lnear one eher Lnear ransformaon A ransformaon A s a lnear ransformaon f sasfes he followng wo properes on ) addon of wo vecors and ) mulplcaon of a vecor by a scalar hese properes mus hold for every vecor n LVS, no jus some vecors Dsrbuve A ( y ) A A y Assocave A ( ) ( A ) Eample: he "radonal," srcly lnear scalar funcons of scalars (e, a s degree polynomal of he followng form whou he consan erm) s a lnear ransformaon f( ) How abou he more general "lnear" scalar funcons of scalars (e, a s degree polynomal of he followng form wh a consan erm )? f( ) Eample: A y n y j = where and y are wo columns of real numbers he rule s defned as: a j j for, n Eample: Roaon of arrows R = or R() =
lmcd Eample: Dfferenaon, Wha s he subjec of hs operaor? In oher words, hs dfferenaon operaor operaes on wha knds of vecors? Arrows? Columns of numbers? Connuous funcons wh s dervaves? D f D( f ) d d f( ) D D f D ( D f ) D f d d d d f( ) d d anoher noaon for a dfferenal operaor We can show ha he ransformaon D sasfes he wo lnear properes hus, D s a lnear ransformaon Dsrbuve D d ( f g) d ( f( ) g( ) ) d d f( ) d d g( ) D f D g Assocave D ( f) f( ) d d ( f( ) ) d d f( ) ( D f) Eample: A more general dfferenal operaor Here s a second-order ODE operaor L d f a( ) d b( ) d d c( ) f( ) a( ) d f( ) d b( ) d d f( ) c( ) y( ) ODE( f ) anoher noaon for a dfferenal operaor Bessel dfferenal operaor: ODE( y ) d d d d y D D y We can show ha he ransformaon ODE sasfes he wo lnear properes hus, ODE s a lnear ransformaon Dsrbuve ODE( f g ) d d d d ( f g ) d d d d f d d d d g ODE f ODE g Assocave ODE( f ) d d d d ( f ) d d Eample: Le he LVS be he space of all polynomals of degree n or less P n ( ) a a a a n n Defne a lnear ransformaon D such ha D P d d f( ) We can show ha D sasfes he wo lnear properes d d ( ODE( f) ) Non-Eample: Wha f he LVS s composed of all real funcons? Answer: No, because he dervave may no es Wha f we ghen he membershp requremen of he LVS so ha he LVS s composed of all real, connuous funcons ha possess he s dervaves? Answer: No, because he resulng funcon may ge kcked ou from he LVS for no havng a s dervave
3 lmcd Eample: Inegraon We can show ha he negral operaor s a lnear one (Wha s he LVS n whch he negral operaor s defned n? In oher words, wha are he membershp requremens?) L f L( f ) f( ) anoher noaon for an negral operaor Wha f he LVS s he space of all polynomals of degree n or less? Answer: No, L s no a lnear ransformaon for hs LVS, because he resulng funcon s a polynomal of degree n+, whch s ecluded from he membershp P n ( ) a a a a n n L P n a P n ( ) a a a n 3 3 n n Wha f he LVS s he space of all polynomals of degree n or less, as before, bu now wh a slghly dfferen negral operaor, lke he one defned below? Answer: Yes, because he resul of he lnear ransformaon now les whn he same LVS Defne L P n P n ( ) a L a P n a a n 3 n Eample: Convoluon negral L f f( ) g(, ) or L f Eample: Laplace ransform L f n s f( ) d F L( f( ) ) F( s ) e Eample: Fourer Sne ransform f( ) g( ) anoher common noaon F f sn( ) f( ) d F( ) f( ) e sn( ) e d Eample: Fourer Cosne ransform F f cos( ) f( ) d F( ) cos( ) e d Eample: Fourer ransform magnary number F f F( f( ) ) F( ) e f( ) d F F( f) F anoher common noaon
4 lmcd Eample: Projecor operaor Consder any real Eucldean space R Le P be a projecor ha projecs a vecor n any Eucldean space ono a fne dmensonal subspace of R y P Noe ha hs s jus a noaon I does no mply mulplcaon of by an operaor P, whch makes no sense he usual algebrac operaons beween wo lnear ransformaons (equaly, addon of wo ransformaons, mulplcaon of a ransformaon by a scalar, mulplcaon of wo ransformaons) are defned n erms of he resul of he ransformaon, never jus he ransformaons Remember, he resul of ransformaon s anoher vecor Srcly speakng, ransformaons all by hemselves has no meanng ransformaons are no objecs/hngs/anmals lke he way vecors are wo lnear ransformaons A and B are equal f for every vecor n he LVS, we have, A B hus, when we speak of wo equal lnear ransformaons, we are really referrng o he wo equal vecors ha resul from applyng he ransformaons o he same vecor he sum/addon of wo lnear ransformaons A and B s defned as C A B f for every vecor n he LVS, we have, C A B he produc/mulplcaon of a ransformaon A by a scalar s defned, f for every vecor n he LVS, we have ( A) A ( ) he produc/mulplcaon of wo lnear ransformaons A and B s defned as C A B f for every vecor n he LVS, we have, C A ( B ) In general, ensor producs are no commuave: ABBA!!! A ( B ) B ( A ) We can demonsrae ABBA wh marces We can also demonsrae hs fac wh any one of he eamples of lnear ransformaons menoned above Eample Change one column of numbers o anoher column of numbers (e, he usual mar) We can easly show ABBA o be rue Eample All polynomals of degree n or less Defne wo lnear ransformaons A and B A P d d P( ) B P P( ) Apply wo lnear ransformaons n wo dfferen sequences A ( B P) d d P( ) P( ) P( ) B ( A d P) P( P( ) d P P( ) P( ) B ( A P ) A ( B P ) ( P( ) P( ) ) P( ) P( ) A ( B P ) B ( A P ) A B B A
Eample All polynomals of degree n or less Defne wo lnear ransformaons A and B 5 lmcd A P d d P( ) B P P( ) Apply wo lnear ransformaons n wo dfferen sequences A ( B d P) P( ) P( ) P( ) d B ( A P ) d P( ) P( ) P( ) dp B ( A P ) A ( B P ) P( ) P( ) A ( B P ) B ( A P ) A B B A
6 lmcd Mar o characerze lnear ransformaon n n-dmensonal LVS If we know how a lnear ransformaon acs on a se of bass vecors, we can easly fnd how acs on any vecors (obvously hey mus belong o he same LVS) In a n-dmensonal LVS, a lnear ransformaon urns a se of n lnearly ndependen bass vecors {,,, } no a new se of n vecors {y,y,,y n } (whch, agan, mus connue o belong o he same LVS) hese oupu vecors from he lnear ransformaon, n urn, can be epressed as a lnear combnaon of he bass vecors {,,, } Any lnear ransformaon A n n-dmensonal LVS can be epressed as a mar A mar (Noe ha he las saemen s rue for a fne dmensonal LVS; s no rue for LVS of an nfny dmenson, eg, all connuous funcons, because we need an nfne number of bass o descrbe a random connuous funcon) Lnear ransformaon on he bass vecors (hs formulaon does no work for -dmenson) A y a a a n A y a a n : a a A y n a n a nn a n In a compac maroaon (some people abhor hs, because we are mng vecors and columns rows of vecors & also vecors ha are recangular-wse arranged se of numbers and marces) hus, he acon of a lnear ransformaon can be characerzed by a mar A mar compac form wh vecors & y lsed n a column A A A y y y n a a a n a a n A A mar compac form wh vecors & y lsed n a row a a a n A A A y y y n a a n A n A mar Here, "A" symbolzes lnear ransformaon; whereas, "A mar " s a mar ha characerzes he lnear ransformaon
7 lmcd Any vecor n he LVS can be represened as a lnear combnaon of he n bass vecors {,,, } n he acon of he lnear ransformaon on an arbrarly chosen vecor s descrbed as y A A n n A A n A n y y n y n y a a n a n y a a a n n n j a j A mar compac maroaon wh vecors & y lsed n a column (agan, remember, some people abhor hs compac noaon because we are mng a column row of vecors wh a column row of scalars) y a a a n y A n y n a A mar y n a n compac maroaon wh vecors & y lsed n a row a a a n y A y y y n a A mar n a n n he choce of bass s no unque he same LVS can be epressed based on anoher se of bass {', ',, ' n }, whch s relaed o he frs se as ' ' mar ' n n ' ' n where mar n ' ' ' n n n nn n n nn ' n Lnear ransformaon on he bass vecors (hs formulaon does no work for -dmenson) A ' y' a' ' a' ' a' n ' n A ' y' a' ' a' ' a' n ' n A: ' n y' n a' n ' a' n ' a' : nn ' n a' ' a' ' a n '
8 lmcd In a compac maroaon wh vecors & y lsed n a column A ' y' a' a' a' n ' a' a' a' n n A ' A ' n y' y' n a' a' n a' a' n a' n a' nn ' ' n a' a' n a' a' n a' n a' nn n n n nn A ' A n n A n a a a n A ' A ' n A n A n n n n nn A A n n n nn a a n a' a' a' n n n a a a n a' a' n a' a' n a' n a' nn n n A' mar mar mar A mar n nn n n In a compac maroaon wh vecors & y lsed n a row n nn a a n If & ' are wo orhonormal ses, hen mar s orhogonal a' a' a' n A ' A ' A ' n y' y' y' n ' ' ' n a' a' a' n a' n a' n a' nn n a' a' a' n n a' n n nn a' n a' a' n a' n a' nn A ' A ' A ' n A n n A n A n n n a a a n n A A A n a n hus, n n nn a n n n nn a a a n n n a' a' a' n a n n a' a' a' n a n n n nn n n nn a' n a' n a' nn
A mar 9 lmcd mar mar A' mar A' mar mar mar A mar smlary ransform equaon compac form wh vecors & y lsed n a column Eample Dervave of polynomals (bass=power seres) All polynomals of degree n or less P() AWe can use y power a seres a f as ahe n n+ bass funcons vecors he dervave operaor acng on hese power seres produce anoher se of polynomals If he LVS s all polynomals of any A degree, here y s a no fne a se a of bass funcons n and a compac noaon does no es D P d A y n d P( y ) n a n compac maroaon wh funcons f lsed n a column f' f' f' f n f' prme f n f f f f n compac maroaon wh funcons f lsed n a row f' f' f' f' n f f f f n f' f prme Any funcon vecor n he LVS (ha s, any polynomals of degree n or less) can be represened as a lnear combnaon of he n+ bass funcons y f f n f n generally acceped D y D f f n f n D f D f n D f n f' f' n f' n compac maroaon wh vecors f n a column f' D y n f' ( prme f ) prme f f' n compac maroaon wh vecors f n a row
D y f' f' f' n f prme f prme lmcd n Eample Dervave of polynomals (bass=legendre polynomals) All polynomals of degree n or less We can use Legendre polynomals P as he n+ bass funcons vecors We fnd he dervave operaor acng on hese power seres produce anoher se of polynomals Noe ha he choce of a se of bass vecors s no unque he represenaon of a gven vecor depends on he choce of bass vecors (see change-of-bass formula) he represenaon of he acon of a lnear ransformaon as a mar depends on he choce of bass vecors and s no unque compac maroaon wh vecors f & P n a column P f A f P P n f P( ) A f( ) P'( ) A prme f( ) A prme A P( ) Pprme P( ) where Pprme A prme A f n compac maroaon wh vecors f & P n a row D P P P P n f f f n A P( ) f( ) A D P P'( ) f prme A P( ) A prme A P( ) Pprme where hus, he mar ha characerzes he acon of dervave on Legendre bass funcons s Pprme A prme A Pprme A prme A Noe ha he same dervave operaor s characerzed by dfferen marces, dependng on he bass funcons vecors Eample Inegraon of polynomals All polynomals P of degree n or less We can use power seres {,,,, } as he n+ bass funcons vecors he negraon operaor acng on hese power seres produce anoher se of polynomals B P P( ) Any members n he LVS (ha s, any polynomals of degree n or less) can be represened as a lnear combnaon of he n+ bass funcons P f f n f n generally acceped B P B f f n f n B f B f n B f n compac maroaon wh vecors f n a column B f B f f f
B f g B f B f n 3 n lmcd f B f f n
compac maroaon wh vecors f n a row B f g B f B f B f B f n f f f f n lmcd Eample Projecon of a gven vecor y n n-dmensonal space ono bass vecor y, P y projecon of y along, Acon of P on each of he bass vecors acon of P on, P y proj, acon of P on, P y proj, : : acon of P on n, n P y projn, compac maroaon wh vecors & y proj n a column,, P P P P y proj y proj y projn,,,, compac maroaon wh vecors & y proj n a row P P P P y proj y proj y projn,,,, 3 n y proj A mar where only one column of A ma s nonzero,,
3 lmcd Any vecor y n he LVS can be represened as a lnear combnaon of he n bass vecors {,,, } y n he acon of he lnear ransformaon on an arbrarly chosen vecor y s descrbed as y proj P y P n n P P n P y proj y proj n y projn y, proj, n, n,,,,, n, n,,, compac maroaon wh vecors y and y proj lsed n a column (agan, remember, some people abhor hs compac noaon because we are mng a column row of vecors wh a column row of scalars) y proj P y n y proj y proj y projn n compac maroaon wh vecors y & y proj lsed n a row,,,,,, A mar y proj P y y proj y proj y projn A mar n,,,,,, If he bass are orhonormal, hen A mar,,,,,,
4 lmcd Eample Projecon of a gven vecor y n n-dmensonal space ono n bass vecors {,,, } y, projecon of y along P y,, n, n,,,, y, projecon of y along P y,, n, n ( F,,,, Pr he pr : : y, projecon of y along n n P n y, n, n, n,,,, Combnng he above n projecon operaors n a column y, P y P y P n y, y,, y,, ( P Ap Pr P y P P P y proj y proj y projn P y P n y P P n P P n P P n n y proj y projn y proj y projn y projn y projnn n,,,,,,,,,,,,,,,,,, n y,,,, y,,,, y,,,, n
5 lmcd y X where y s a column of scalar producs beween y and he bass vecors {,,, } and X s a mar of scalar producs of he bass vecors X y y, If he bass vecors are orhogonal, X dagonal, If he bass vecors are orhonormal, X I mar y, Combnng he above n projecon operaors n a row y, P y P y P y, y, n y,,, P y P y P n y n P P P P P P P n P n P n y proj y proj y projn n y proj y proj y projn y projn y projn y projnn,,,,, n,,,,,,,,,,,,,,,, y, y, y, n n,,,,,, y X where y s a row of scalar producs beween y and he bass vecors {,,, } and X s a mar of scalar producs of he bass vecors
6 lmcd Eample Projecon of a gven vecor y n n-dmensonal space ono anoher vecor {,,, } P ( y, ) y projecon of y along (, ) Any vecor y n n-dmensonal LVS can be represened as a lnear combnaon of he bass vecors y n n P ( y, ) y n, n (, ) n, n n Addon sum of wo ransformaons A & B ( A B) A B C A B C mar A mar B mar Mulplcaon of a ransform A by a scalar A ( A ) C A C mar A mar Mulple lnear ransformaons n sequence: B followed by A A ( B ) ( A B) C C A B C mar A mar B mar A followed by B B ( A ) ( B A) D D B A D mar B mar A mar In general, he order of operaon maers: A B B A A mar B mar B mar A mar
7 lmcd Ideny ransformaon An deny ransformaon s a ransformaon ha does no modfy he gven vecor, e, smply do nohng! I hus, he acon of he deny ransformaon on n lnearly ndependen bass vecors {,,, } s compacly epressed as: (Agan, remember, some people abhor hs compac noaon) Rule for deny ransformaon: y I y I y n I compac maroaon wh vecors n a column y I y y y n I mar compac maroaon wh vecors n a row I y y y y n I mar Here, "I" symbolzes deny ransformaon; whereas, "I mar " s an deny mar ha characerzes he deny ransformaon Inverse ransformaon Gven a lnear ransformaon A n a LVS, an nverse ransformaon reverses he ransformaon performed on any vecor by A In oher words, he lnear ransformaon B s an nverse of he ransformaon A f, for every vecor n he LVS, we have, A ( B ) B ( A ) Noe ha boh relaonshps have o hold, as ABBA n general A B B A I If B s an nverse of A, hen A s an nverse of B We commonly use he superscrp " - " o denoe he nverse ransformaon of A: B or A - If nverse ess, s unque If a lnear ransformaon A has an nverse, s sad o be non-sngular Oherwse, s sngular Agan, hs ermnology s a very general one We do no even have o be concerned abou a real Eucldean space, fne dmenson, mar represenaon, ec If A has an nverse, hen A A y mples y In fac, hs s a necessary condon for A o be non-sngular In plan Englsh, gven wo dencal vecors z & z ha resul from he same lnear ransformaon A, (z =A and z =Ay), f we race back he lnear ransform by akng B=A - o see where each has orgnaed from, we should end up a eacly he same sarng vecor (meanng =y) A A A y n y y y n a a a n a a n A y A y A y n a a a n a a n y y y n
8 lmcd A common ms-concepon When we reverse he vecor drecon, we urn a vecor no a reversed nvered (e negave) vecor, and we denoe hs wh a mnus sgn: - he process of reversng vecor drecon s no an nverse ransform Conversely, an nverse ransformaon s no "-A" he noaon "-A" all by self has no meanng; "-A", whch s a common abbrevaon for "-(A)" mahemacally means he negave of he oupu vecor " A", jus as -f(), whch s an abbrevaon for "-(f())", no "(-f)()", means he negave of he oupu from he funcon f Eample Roaons of arrows have nverses (e, he orgnal roaon can be undone) Eample Column of numbers (eg, a pon vecor n Caresan coordnae) Do no confuse " " here (whch s a number) wh " " n he prevous pages (whch s a vecor) n Defne A y n he usual way y a a a n = Acon of ransformaon A epressed n a compac maroaon y a a a n a j j for, n a a a n A y y a A mar A mar a y n a n a n he frs "" s lnear operaon; he second "" s mar mulplcaon We can hnk of ec as one of he bass vecors (raher han a number embedded n a column compac maroaon wh vecors n a column (has he same form as he he lnear algebrac equaon bu & y n each column represens a bass vecor) y a a a n A y y a A mar y n a n compac maroaon wh vecors n a row a a a n A y y y y n a A mar a n
9 lmcd he lnear ransformaon A has nverse only f we can fnd from gven y hus, A s non-sngular f and only f de(a mar ) Do no confuse " " here (whch s a number) wh Eample Row of numbers " " n he prevous pages (whch s a vecor) n Defne A y n he non-usual way y a a a n a j j for, n In compac maroaon = a a a n A y y y y n a A mar a n Eample Projecon of arrows has no nverse (e, sngular) Reason: Many arrows yeld he same projecon; hus, we canno ge back he orgnal arrow Eample All polynomals of degree n or less Defne wo lnear ransformaons A and B A P d d ( P( ) ) B P P( ) Apply wo lnear ransformaons n sequence A ( B P) d d P( ) d d P( ) P( ) B ( A P ) d ( P( ) ) d( P( ) ) ( P( ) ) P( ) Snce A ( B P ) B ( A P) P he above wo lnear ransformaons A and B are nverse of each oher Eample: Laplace ransform (Snce he LVS s composed of connuous funcons; he dmenson s nfny, and here s no compac mar represenaon of he acon of he ransformaon on bass vecors) A f( ) s f( ) d F( s) e A F( s ) s F( s ) ds f( ) e Co F d r F
Eample: Fourer Sne ransform F f sn( ) f( ) d F( ) F F( ) Eample: Fourer Cosne ransform F f Eample: Fourer ransform F f( ) F F( ) cos( ) f( ) d F( ) F F( ) e f( ) d F( ) e F( ) f( ) lmcd sn( ) F( ) f( ) cos( ) F( ) f( ) F Null Space he null space of a lnear ransformaon A s he se of of all vecors n he LVS such ha A Naurally, hese specal se of vecors le whn he same LVS n whch he lnear ransformaon A s defned hus, he null space s a subspace of he LVS Noe ha A does no need o have an nverse n order for us o speak of he null space re sp D g fro ( F ( F An G bu
lmcd Orhogonal ransformaon An orhogonal ransformaon s a lnear ransformaon A ha acs on a vecor and does no change s lengh In oher words, s a ransformaon ha merely "roae" a vecor (hus, change s angle bu no s lengh) A common ms-concepon: an orhogonal ransformaon does no mean he npu vecor and he oupu vecor y=a are orhogonal o each oher! In oher words, he angle of "roaon" beween he npu vecor and he ou vecor y=a does no need o be 9 degrees Anoher common ms-concepon: do no confuse orhogonal vecors wh orhogonal lnear ransformaons! Bass vecors ha we use o descrbe a gven vecor n LVS do no need o be orhogonal hus, we can alk abou orhogonal ransformaons n LVS ha s represened by non-orhogonal bass vecors Conversely, n a LVS represened by orhogonal bass vecors, he lnear ransformaon can be anyhng (ceranly no lmed by orhogonal ransformaons) Snce we are referrng o vecor lengh, we need o defne a scalar produc and we are workng wh real Eucldean space defnon of orhogonal ransformaon A scalar produc afer ransformaon = scalar produc before ransformaon ( A, A ) (, ) "Orhogonal" s a way o descrbe he "before" and he "afer" of a lnear ransformaon A on An orhogonal ransformaon preserves scalar produc (no jus for, bu for any & y pars) ( A, A y ) (, y) Proof Le z be he sum of wo vecors & y z y Apply ransformaon A o z; snce A s orhogonal, by defnon of A beng "orhogonal" we have for z ( A z, A z ) ( z, z) We subsue z no he above defnon ( A ( y ), A ( y ) ) ( ( y ), ( y) ) ( A, A ) ( A, A y ) ( A y, A ) ( A y, A y ) (, ) (, y ) ( y, ) ( y, y) Agan, by defnon of A beng "orhogonal" we have for & y: ( A, A ) (, ) ( A y, A y ) ( y, y) (, ) ( A, A y ) ( y, y ) (, ) (, y ) ( y, y) Comparng he LHS & RHS, we have ( A, A y ) (, y) Acon of A on each of he bass vecors {,,, } acon of A on A y a a a n acon of A on : A y a a a n : acon of A on A y n a n a n a nn Acon of ransformaon A epressed n a compac maroaon wh vecors & y n a column A A A A y A mar y y y y n a a a n a a n A mar A mar a a a n a a n An op ev me
lmcd compac maroaon wh vecors & y n a row a a a n A A A A y y y n a a n Mar of varous scalar producs beween bass vecors (A,A j )=(, j ) A, A A, A A, A a a a n,,, a a A, A A, A A, A A, A A, A A, A a a n,,,,,, a a n,,,,,,,,, A mar X A mar X where X s a mar of scalar producs of he bass vecors If A s an orhogonal lnear ransformaon, by defnon of "orhogonal", he above relaonshp holds; follows he smlary ransform equaon beween A mar and he nverse of s ranspose, (A mar ) -, wh he mar of scalar producs of bass X as he smlary ransform mar A mar X X A mar A mar A mar for general bass vecors ha are no necessarly orhogonal (bass need o be lnearly ndependen, bu do no need o be orhogonal) for orhonormal bass vecors {,,, } X I mar A mar A mar or equvalenly A mar A mar A mar A mar I mar As shown above, he columns n A mar are orhogonal (A mar A mar =I mar ) ; he rows n A mar oo are orhogonal (A mar A mar =I mar ) (Bu each column n A mar s no he same ype of vecor ha he lnear ransformaon A acs on -- agan, hs s a consan source of confuson, and many are agans he compac "mar" noaon) Eample Roaon n a plane wh Defne a lnear ransformaon A A Defne scalar produc (, y ) y y ( ) cos( ) sn( ) sn( ) cos( ) ( A ) cos( ) sn( ) sn( ) cos( ) hus, A s an orhogonal ransformaon Show ha he scalar produc s preserved ( )
( A, A y) 3 lmcd cos( ) sn( ) cos( ) y sn( ) y + sn( ) cos( ) sn( ) y cos( ) y y y (, y) Symmerc ransformaon A symmerc ransformaon s a lnear ransformaon A wh he followng propery for every par of vecors and y n real Eucldean space defnon of symmerc ransformaon ( A, y ) (, A y) scalar produc afer A acs on = scalar produc afer A acs on y In oher words, gven any wo vecors and y, swappng he operaon, wheher A acs on or on y, does no change he resulng scalar produc beween hem, meanng he relaonshp (eg, angle) A common ms-concepon: a symmerc ransformaon does no mean he npu vecor s symmerc, nor he oupu from he ransformaon y=a s symmerc! Anoher common ms-concepon: do no confuse orhogonal bass vecors wh symmerc lnear ransformaons! Bass vecors ha we use o descrbe a gven vecor n LVS do no need o be orhogonal; hey can be anyhng as long as hey are lnearly ndependen hus, we can alk abou symmerc ransformaons n LVS ha s represened by non-orhogonal bass vecors Snce we are referrng o scalar produc, he same ransformaon can be symmerc wh respec o one scalar produc defnon, bu no symmer wh a dfferen scalar produc defnon hs s analogous o he orhogonaly beween wo vecors: wo same vecors are orhogonal wh one scalar produc defnon, bu non-orhogonal wh a dfferen scalar produc defnon Bascally, s an "equal opporuny" ransform "Symmercal" s a way o descrbe ha applyng lnear ransformaon Ao a vecor, n erms of s relaonshp o all he oher vecors y as quanfed by he scalar produc (A,y), does no depend on whch vecor A acs on Had A aced on a dfferen vecor y, he resulng scalar produc (Ay,) would have been dencal o acng on, e, (A,y) he change brough abou by A acng on a random vecor s ndependen of he vecor ha we pu hrough A Each and every vecor s capable of causng he same resul "Orhogonal" deals wh "before" and "afer"; whereas, "symmercal" deals wh laerally swappng an npu vecor wh anoher npu vecor y Acon of A on each of he bass vecors {,,, } acon of A on A y a a a n acon of A on : A y a a a n : acon of A on A y n a n a n a nn Acon of ransformaon A epressed n a compac maroaon wh vecors & y n columns A A A A y A mar y y y y n a a a n a a n A mar A mar a a a n a a n
4 lmcd compac maroaon wh vecors & y n rows a a a n A A A A y y y n a a n Mar of varous scalar producs beween bass vecors and he oupu of A on bass vecors (A, j )=(,A j ) A, A, A,, A, A, A A, A, A,, A, A, A X A, A, A,, A, A, A A mar a a a n A mar X a,,, a n,,,,,,,,,,,, a a a n a,,, where X s a mar of scalar producs of he bass vecors If A s a symmerc lnear ransformaon, by defnon of "symmerc", he above wo equaons are equal he resulng relaonshp follows he smlary ransform equaon beween A mar and s ranspose A mar, wh he mar of scalar producs of bass X as he smlary ransform mar A mar X X A mar for general bass vecors ha are no necessarly orhogonal (bass need o be lnearly ndependen, bu do no need o be orhogonal) A mar A mar for orhonormal bass vecors {,,, } X I mar a n a j a j Eample columns of numbers Defne a lnear ransformaon A A a a a a Defne scalar produc (, y ) y y ( A, y ) (, A y ) requremen for a symmerc ransformaon a a y a y a y a y a y y a a y y hus, he requremen for a symmercal ransformaon s a a
5 lmcd Eample Convoluon Inegral Le he LVS be all connuous funcons n =[, ] Defne a lnear ransformaon A (convoluon negral), where K s a connuous funcon A f K(, ) f( ) K Defne scalar produc ( f, g) f( ) g( ) ( A f, g ) ( f, A g ) requremen for a symmerc ransformaon K K(, ) f( ) g( ) d f( ) K(, ) g( ) K(, ) f( ) g( ) d K(, ) f( ) g( ) d he ( K(, ) K(, ) ) f( ) g( ) d F hus, he requremen for a symmercal ransformaon s K(, ) K(, ) Skew-Symmerc ransformaon A skew-symmerc ransformaon s a lnear ransformaon A wh he followng propery for every par of vecors and y n real Eucldean space ( A, y ) (, A y) compac maroaon A mar X X A mar A mar for general bass vecors no necessarly orhogonal A mar for orhonormal bass vecors {,,, } d In K a j a j
6 lmcd ranspose Any lnear ransformaon A n n-dmensonal LVS has a unque ranspose he ranspose of a lnear ransformaon A has he followng propery for every par of vecors and y n real Eucldean space B s he ranspose of A, and A s he ranspose of B ("he", and no "a", means ha he ranspose of A ess and s unque) Noaon: A ( A, y ) (, B y ) ( A, y ), A y Specal case f A s a symmercal lnear ransformaon, by defnon we have, ( A, y ) (, A y) A A for symmercal lnear ransformaon Some properes of ranspose (ncludng, bu no lmed o, he radonal "mar" ype) ( A B) A ( A ) A ( A B ) B A B Acon of A on each of he bass vecors {,,, } acon of A on A y a a a n acon of A on : A y a a a n : acon of A on A y n a n a n a nn Acon of ransformaon A epressed n a compac maroaon wh vecors & y n a column A A A A y A mar y y y y n a a a n a a n compac maroaon wh vecors & y n a row A mar A mar a a a a a n a a n a n A A A A y y y n Mar of varous scalar producs beween bass vecors (A, j )=(,B j ) A, A, A, A, A, A, A, A, A, a a a n A mar X a a n a a n,,,,,,,,,
7 lmcd, A, A, A,,, a' a' a' n, A, A, A, A, A,,,, A,, n X A' mar X B mar b j a' j where X s a mar of scalar producs of he bass vecors and, B mar A' mar descrbes he acon of ranspose of A A =B on each of he bass vecors {,,, } acon of A =B on B y' a' a' a' n b b b n acon of A =B on : B y' a' a' a' n b b b n : acon of A =B on B y' n a' n a' n a' nn b n b n b nn A mar X X A' mar X B mar A mar a j a' j b j a' a' n a' a' n for general bass vecors ha are no necessarly orhogonal (bass need o be lnearly ndependen, bu do no need o be orhogonal) B mar for orhonormal bass vecors {,,, } X I mar Because we can always fnd an orhonormal se of bass vecors n n-dmensonal space (Gram-Schmd assures us ha hs can always be done), a lnear ransform A can ac on he bass {,,, }, hen we can always defne anoher lnear ransform B=A ha acs on he bass {,,, } accordng o he above rule hus, rrespecve of he lnear ransformaon (no need o be orhogonal or symmerc, ec), here s always a unque ranspose B=A a' n a' nn