Analytic Geometry. ) of numbers where P are the x coordinate and y coordinate of P.

Transcription

1 Chpter lytic Geoetry lytic geoetry seres s ridge etwee lger d geoetry tht kes it possile for geoetric proles to e soled y es of lger or ice ers The ojects i geoetry icldes lies ples cres d srfces d the ojects stdied i lger re rios differet opertio systes eqtios etc Geoetry lytic Geoetry lger Coordite Systes Defiitio (Crtesi Ple) Tke two perpediclr itersectig stright lies d estlish scle o ech lie oe is clled the -is the other is clled the y -is The poit of itersectio is the origi The there is correspodece etwee poits P d pirs ( P P y ) of ers where P d Py re the coordite d y coordite of P Rerk () The -is is geerlly chose s horizotl lie with positie coordites to the right of the origi d the y -is is erticl lie with the positie coordites oe to the origi () The scle is sed o oth es Theore The distce etwee two poits P ( y ) d ( y ) Q is ( ) ( y ) PQ y

2 ple 4 If P ( ) ( ) Defiitio 5 Q d PQ fid The slope of lie is the tget of the iclitio; ths where θ is the iclitio of the lie tθ Theore 6 lie throgh P ( ) d ( ) y P y where y y hs slope Theore 7 If l d l re o-perpediclr lies with slopes d respectiely d α is y gle fro l to l the tα Grphs of qtios Defiitio The grph of eqtio ( y) f is the set {( y) : f ( y) } ple () The grph of the eqtio y is lie () The grph of the eqtio y is circle with cetre t origi d rdis () More eples qtios y c Grphs ie y llipse

3 y Hyperol 4cy y 4c Prol Qestio: Wht is the grph of the eqtio y cy d ey f? Rerk: llipses prols d hyperols re clled coic sectios ie cres fored y the itersectio of ple with right circlr coe Trsfortio of Coordites To represet poits i the ple lgericlly we choose two perpediclr scled lies these lies for coordite syste Sch syste is ot iqe If we re gie two coordite systes the poit i the ple correspods two coordites d therefore grph y he two differet eqtios The eqtio of grph y ecoe ssttilly siple if the coordite syste is properly chose y coordite syste c e otied y oig fied syste There re two sic oe of coordite systes: Trsltio rottio ery oe is coitio of the sic oe Propositio Sppose the coordite es re trslted to get ew syste i sch wy tht the origi of the ew syste is the poit of the old syste For y poit P i the ple sppose ( y) y re the coordites of P i the old d ew syste respectiely the we he d ( ) h y y k ple Deterie the grph of the eqtio y 4 y 4 Rerk For y eqtio By Cy D y F we c fid trsltio sch tht the first degree ters re reoed To reoe the cross ter we eed to se rottio Propositio 4 (qtio of Rottio) Sppose the es of gie syste re rotted throgh gle θ the the coordites of poit P he the followig reltios:

4 cosθ y siθ y siθ y cosθ ple 5 Idetify the grph of the eqtio 4y y 6 Theore 6 If the eqtio is trsfored ito the eqtio the Tht is By Cy D y F B y C y D y F B 4C B 4 C B 4C is irit der trsfortio Theore 7 The eqtio By Cy D y F represets hyperol ellipse or prol whe egtie B 4C is positie zero or

5 Chpter ier Systes d Mtrices ier Systes d Soltios ple I the clide ple the eqtio of lie hs the for y c d c re costts Sppose lie psses throgh the two poits ( ) where P ( 7) Fid the eqtio of this lie P d et ly k e the eqtio of the lie We eed to deterie the les of l d k Sice the gie poits re o the lie so their coordites stisfy the eqtio ie we he the followig lier eqtios l k 7l k The prole ow ecoes to fid grop of les for eqtios re stisfied l d k sch tht the two Defiitio lier syste i riles (kows) is grop of lier eqtios M where ij s d i s re costts Defiitio () soltio of lier syste is grop of les of kows k k k which stisfy ech of the lier eqtios i the syste

6 () We cll lier syste cosistet if it hs t lest oe soltio otherwise it is clled icosistet () lier syste is clled hoogeeos lier syste if ll the costts o the right side of the eqtios re zero Theore 4 ch lier syste hs either o soltio iqe soltio or ifiite y soltios ple 5 Soe specil types of lier systes d their soltios () The followig lier syste oiosly hs ectly oe soltio: tht is Geerlly the followig lier syste hs ectly oe soltio: M () The followig lier syste hs ifiitely y soltios: Gie y le to we get soltio et this lier syste c e writte s s s The the geerl soltio of Qestio: Gie ritrry lier syste c we trsfor ito lier syste i the specil for siilr to () or () so tht d he the se soltios s s Theore 6 (Three leetry Opertios) The followig opertios o lier systes do ot chge soltios: () Mltiply the two sides of eqtio y o-zero er () Iterchge two eqtios () dd ltiple of oe eqtio to other

7 ple 7 Trsfor the followig syste ito its specil for y sig the three eleetry opertios the fid the soltios: geted Mtri of ier Syste For lier syste if we kow the kows the the key fctor which deterie the soltios re the coefficiets ij d the costts i It wold e ch sipler if we drop kows d keep ij d i t their positio I this wy we get rectglr of ers which will e clled the geted tri of the syste We c lso write ck the syste ccordig to its geted tri Defiitio The geted tri of the followig lier syste M is the rectglr rry of ers We cll this tri ple () Write ot the geted tri of the followig lier syste () Write dow the lier syste of which the geted tri is

8 4 4 sse tht the kows re d y The three eleetry opertios o lier syste correspod to the followig three eleetry row opertios o its geted tri Defiitio (leetry Row Opertios o Mtrices) For tri the followig re clled eleetry row opertios o : () Mltiply row throgh y o-zero er k ( ) kr i () Iterchge two rows ( r i r j ) () dd ltiple of oe row to other ( r i kr j ) The geted tri of lier syste i specil for he soe specil fetres these trices re clled redced row-echelo for Defiitio 4 (Row-chelo For d Redced Row-chelo For) Cosider the followig properties of tri: () If row does ot cosist etirely of zeros the the first o-zero er i the row is (clled ledig ) () ll the zero row st e groped together t the otto of the tri () I y two sccessie rows tht do ot cosist etirely of zeros the ledig i the lower row st occr frther to the right th the ledig i the higher row (4) ch col tht cotis ledig hs zeros eerywhere tri is sid to e i row-echelo for if it stisfies coditios () () d () If it stisfies ll the for coditios it is sid to e i redced row-echelo for ple 5 Deterie which of the followig trices re i redced row-echelo for or row-echelo for B C D

9 ple 6 Sppose tht the geted tri for lier syste i riles d hs ee redced to the gie redced row-echelo for Sole the syste Soltio The correspodig lier syste is et s the the geerl soltio of the gie lier syste is s s s Qestio: For gie tri is it lwys possile to trsfor ito (redced) rowechelo for y sig eleetry row opertios? If the swer is yes the how to redce tri ito row-echelo for d redced row-echelo for? Gssi liitio d Gss-Jord liitio Theore () ery tri c e trsfored ito row-echelo for s well s ito redced row-echelo for y sig the eleetry row opertios () ery tri c e trsfored ito iqe redced row-echelo for lgorith Cosider the followig trsfortios o gie tri (Step ) octe the leftost col tht does ot cosist etirely of zeros (Step ) Iterchge the top row with other row if ecessry to rig ozero etry to the top of the col fod i step (Step ) If the etry tht is t the top of the col fod i step is ltiplyig the first row y to get ledig 4 (Step 4) dd sitle ltiples of the top row to the rows elow so tht ll etries elow the ledig ecoe zero

10 5 (Step 5) Now coer the top row i the tri d egi gi with Step pplied to the s tri tht reis Cotie i this wy til the etire tri is i row-echelo for 6 (Step 6) Begiig with the lst o-zero row d workig pper wrd dd sitle ltiples of ech row to the rows oe to itrodce zeros oe the ledig s Usig steps -5 eery tri c e trsfored ito row-echelo for d this procedre is clled Gssi eliitio Usig ll the 6 steps eery tri c e trsfored ito its redced row-echelo for d this procedre is clled Gss-Jord eliitio ple Trsfor the followig tri ito row-echelo for: Soltio Step octe the leftost col tht does ot cosist etirely of zeros Step Iterchge the top row with other row if ecessry to rig o-zero etry to the top of the col fod i Step Step If the etry tht is t the top of the col fod i Step is ltiplyig the first row y to get ledig Step 4 dd sitle ltiples of the top row to the rows elow so tht ll etries elow the ledig ecoe zero

11 Step 5 Now coer the top row i the tri d egi gi with Step pplied to the s tri tht reis Cotie i this wy til the etire tri is i row-echelo for This tri is ow i row-echelo for Step 6 Begiig with the lst o-zero row d workig pwrd dd sitle ltiples of ech row to the rows oe to itrodce zeros oe the ledig s Rerk 4 () If the geted tri of lier syste is i redced row-echelo for the the soltios c e otied y osertio If the geted tri is i rowechelo for we c sole the syste y ck sstittio () To sole lier syste y es of Gssi eliitio we first write dow the geted tri of the syste The trsfor ito row-echelo for B The write dow the lier syste correspodig to B t lst sole the ew syste to oti the soltios which re lso the soltios of the origil lier syste I this cse we sy tht we sole the lier syste y Gssi eliitio If the tri is redced to its redced row-echelo for we sy tht we sole the lier syste y sig Gss-Jord eliitio

12 Gssi eliitio ier syste geted tri Row-echelo for ier syste Soltios Gss-Jord eliitio ier syste geted tri Redced row-echelo for ier syste Soltios ple 5 Sole the followig lier syste y Gss-Jord eliitio Soltio The geted tri for the syste is The redced row-echelo for is The correspodig lier syste ecoes So the soltio is Theore 6 () ery hoogeeos lier syste hs t lest oe zero soltio which is clled the triil soltio () hoogeos lier syste tht hs ore er of kows th eqtios hs ifiitely y soltios

13 Rerk 7 () Not eery lier syste hs soltio () Gie y lier syste there is correspodig hoogeeos syste which is otied y lettig ll the costts o the right sides e zero d leig others chged There is close reltio etwee the soltios of d the soltios of This will e discssed i lter chpters ple 8 Sole the followig lier syste: Soltio The geted tri of the lier syste is The redced row-echelo for is The correspodig syste is The geerl soltio is t t s t s Mtrices d Opertios o Mtrices

14 I the lst sessio we he see tht trices re ery sefl i solig lier systes Mtrices lso rise i y cotets other th s geted trices for lier systes I this sectio we stdy systeticlly the lgeric opertios o trices Defiitios d ples Defiitio 4 tri is rectglr rry of ers The ers i the rry re clled the etries i the tri Two trices re eql if they he ectly the se size d se etries Rerk 4 () tri B tht hs rows d cols is clled tri The size of tri is defied to e () oe col (row) tri is clled col (row resp) tri () geerl tri of size is of the for The trices [ ][ ] [ ] re clled the row trices of d the trices M M re clled the col trices of M (4) et the etry of i ith row d jth col e ij the the tri is lso The etry of t ith row d jth writte s [ ij ] or jst [ ij ] col is lso deoted y ij (5) tri is clled sqre tri of order The etries re sid to e o the i digol of

15 ple 4 [ ] [] 9 d c Opertios o Mtrices Defiitio 44 (dditio) If [ ] ij d [ ] ij B re two trices with the se size the the dditio B of d B is lso tri d [ ] ij ij B The differece B of d B is [α ] ij ij B ple 45 et C B Clclte B B B Is C defied? Wht is ( ) ( ) B B Is B B?

16 Defiitio 46 (Sclr Mltiplictio) If [ ij ] is y tri d c is y sclr the c is the tri [ c ij ] c ( )B is slly writte s B ple 47 et the c c c c c Defiitio 48 (Prodct) If [ ij ] r is r defied to e the tri d B [ ij ] r is r tri the the prodct B is tri whose etry t i th row d j th col is i j i j irrj ple 49 Cosider the trices 9 9 C Decide which of B B C d C re defied I cse of defile clclte the prodct Propositio 4 et [ ij ] r d B [ ij ] r e two trices The (i) the j th col tri of B [ jth col tri of B] B [ ith row tri of ]B d i th row of (ii) If deote the row trice of d deote the col trices of B the d B [ ] [ ]

17 B B B B B M M ple 4 (Mtri For of ier Syste) Cosider M This lier syste is eqilet to the eqtio of trices M M If we deote these trice y d the the tri eqtio is clled the tri for of the lier syste is clled the coefficiet tri of the syste Defiitio 4 (Trspose) et e tri the the trspose of deoted y T is the tri otied y iterchgig the rows d cols of So if [ ] ij the [ ] ji T If the

18 T Defiitio 4 (Trce) The trce of sqre tri is the dditio of ll etries o the i digol If tr the ( ) Properties of Opertios o Mtrices Propositio 5 ssig tht the sizes of trices re sch tht the idicted opertios c e perfored the the followig eqtios hold: () B B () ( B ) C ( B C ) (c) ( BC) ( B)C (d) ( B C ) B C ( B C ) B C (e) ( B ) B (f) ( ) (g) ( ) ( ) (h) ( B) ( ) B ( B) (i) B ( )B For y two positie itegers d let e the tri whose etries re zero is clled the zero tri with size We lso se to deote zero tri sqre tri of which ll the etries o the i digol re d the rest of the etries re zero is clled idetity tri We se I or jst I to deote the idetity tri of size

19 Theore 5 () () (c) (d) (e) If is y tri The I I Defiitio 5 sqre tri is iertile if there is sqre tri B of the se size sch tht B B I I this cse B is clled the ierse of ple () B is ierse of () 5 is ot iertile (Why?) Theore 55 () If B d C re ierses of the B C ; () If d B re iertile the so is B B B () If d B re iertile the ( ) er of trices This is tre for y fiite (4) If is iertile the for y o-zero sclr k k is iertile d ( ) k k By () the ierse of iertile tri is iqe Theore 56 T T () ( ) T T T () ( ) B B T T () ( k ) k

20 (4) ( ) T T T B B (5) If is iertile the so is T d ( ) ( ) T T 6 leetry Mtrices d Method of Fidig leetry Mtrices ple 6 Cosider the followig three trices: k k d re otied fro the idetity tri y sig sigle eleetry row opertio et d c The k d k c d c k k d c ch of the trices d is lso otied y sig sigle eleetry row opertio to the tri Defiitio 6 tri is clled eleetry tri if it c e otied fro the tri I y sig sigle eleetry row opertio ple 6 The followig re ll eleetry trices: 9 Bt

21 4 is ot eleetry tri Why? Theore 64 If the eleetry tri reslts fro perforig row opertio p o I d if is tri the is the tri tht is otied y perforig the se opertio p o ple 65 Cosider 8 Perfor the opertio 8r r o the copre the reslt with leetry Mtrices d the Ierse of Mtri Theore 66 ery eleetry tri is iertile The ierse of eleetry tri is still eleetry tri ple 67 et k k The k k Theore 68 If is tri the the followig re eqilet: () is iertile; () X hs oly the triil soltio; (c) the redced row-echelo for of is I ;

22 (d) c e epressed s prodct of eleetry trices ple 69 press s prodct of eleetry trices where Soltio The tri c e redced to its redced row-echelo for s follows: ( ) ( ) r r r r r The eleetry trices correspodig to ech of the opertios re Ths I d Method of Clcltig If I k k with i eleetry trices the I k k This idictes tht if we se eleetry row opertios k p p p to redce to I the sig the se opertios o I we get Hece to oti we pt d I side y side to for lrger tri The pply eleetry row opertios to the ew tri to trsfor the s tri ito the idetity tri The s tri I will the e trsfored ito [ ] [ ] I I k p p ple 6 Fid where Soltio [ ] I So

23 ple 6 Sole the followig lier syste y y Soltio The tri for of the lier syste is where y X Sice is iertile so the iqe soltio is 4 X

24 Chpter Deterits Deterits of Sqre Mtrices ple et c d We kow tht if d c the c e redced ito the idetity tri y eleetry row opertio so is iertile The er d c is clled the deterit of d is deoted y det ( ) I the followig we shll defie er det ( ) for eery sqre tri d iestigte the properties d pplictios of this fctio Defiitio () perttio ( j j j ) of the set { } is rrgeet of these itegers i soe order withot oissios or repet () ierse is sid to occr i perttio ( j j ) iteger precedes sller oe j wheeer lrger (c) perttio is clled ee if the totl er of iersios is ee er Otherwise it is clled odd perttio ple ( 4 5 ) is odd perttio of { 4 5 } ( 4 5 ) is ee perttio of { 4 5 } Defiitio 4 defie ij For ech sqre tri ( ) where i the s j j j otherwise it tkes egtie sig ( ) det ± j j i j tkes positie sig if ( j j ) j is ee perttio

25 will lso e sed to deote the deterit of ple () the det ( ) 4 ( ) 5 9 Geerlly if () det ( I ) the ( ) for ll it tri det () (4) ltig Deterits y Row Redctio Theore et e sqre tri () If hs row (col) of zeros the det ( ) () det ( ) T det( ) (c) If is i digol ie triglr tri the det ( ) ( ) det is the prodct of ll etries o the Qestio: Sppose tri is otied y perforig sigle row opertio o sqre det? tri the wht is the reltio etwee ( ) det d ( ) Theore et e sqre tri i () If kr B the ( B ) k det( ) det

26 () If B j i kr r the ( ) ( ) B det det () If B j i r r the ( ) ( ) B det det Hece i i i i i i k k k k i i i j i j i j i k k k i i i j j j j j j i i i ple 6 5 The ( ) ( ) ( ) ( ) det det 5 5det det I I d ( ) ( ) det det I Propositio 4 If is sqre tri with two proportiol rows or two proportiol cols the ( ) det

27 ple 5 (ltig Deterits y Row Redctio) det More Properties of the Deterit Fctio Theore et B d C e trices tht differ oly i r th row d the r th row of C c e otied y ddig correspodig etries i the r th row of d B The ( ) ( ) ( ) B C det det det The se reslt holds for cols r r r r r r

28 r r r r r r j j j j j j j j j j j j Theore If d B re sqre trices of the se size the ( ) ( ) ( ) B B det det det Theore sqre tri is iertile if d oly if ( ) det ple 4 Deterie for which les of k the followig tri is iertile: 5 k k Theore 5 If is tri the the followig coditios re eqilet: () is iertile () hs oly the triil soltio (c) The redced row-echelo for of is the idetity tri I (d) c e epressed s prodct of eleetry trices

29 (e) (f) is cosistet for eery tri hs iqe soltio for eery tri (g) det( ) 4 Cofctor psio Defiitio 4 et e sqre tri () The ior of etry ij deoted y M ij is defied to e the deterit of the s tri tht reis fter the i th row d j th col re deleted fro () For ech etry ij i j C ij M is clled the cofctor of ij of ( ) ij ple 4 () et () et c d Fid C d C Clclte C C Fid C C d C ple 4 et The det ( ) ( ) ( ) ( ) C C C

30 I geerl we he the followig theore Theore 44 The deterit of sqre tri c e otied y ltiplyig the etries i y row (or col) y their cofctors d ddig the resltig prodcts Cofctor epsio log ith row det C C C ( ) i i i i i i Cofctor epsio log jth col det C C djoit of tri ( ) j j j j jcj Defiitio 45 et e tri d C ij e the cofctors of ij the the tri C C M C C C C M is clled the tri of cofctors fro The trspose of this tri is clled the djoit dj of d is deoted y ( ) C C C M Theore 46 If is iertile tri the det ( ) dj ( ) ple 47 Fid the ierse of Soltio dj ( ) 4 det ( ) 6

31 so Theore 48 (Crer s Rle) is lier syste with kows d eqtios sch tht det( ) If the syste hs iqe soltio: where ech ( ) ( ) ( ) ( ) ( ) ( ) det det det det det det i is the tri otied y replcig the i th col of y the ple 49 Use the Crer s rle to sole the followig lier syste: z 6 4y 6z y z 8 Soltio ( ) 44 det( ) 4 det( ) 7 det( ) 5 det 8 8 So the soltio is 4 6 8

32 Chpter 4 Vectors i -Spce d -Spce 4 Geoetric Vectors d Opertios The physicl qtities c e clssified ito two types: () Sclrs: re legth tepertre weight etc () Vectors: elocity force displceet ch of the first type of qtities c e deteried y sigle er Bt secod type qtity c ot e descried y sig oly er Rerk 4 () ector c e represeted geoetriclly s directed lie seget or rrow i -spce or -spce; the directio of the rrow specifies the directio of the ector d the legth of the rrow defies its gitde () The til of the rrow is clled the iitil poit of the ector d the tip of the rrow the teril poit () Vectors with the se directio d se legth re clled eqilet We regrd eqilet ectors s the se (4) The ector of legth zero is clled the zero ector d is deoted y Defiitio 4 (Opertios o Vectors) Gie two ectors d w () The s w is defied to e the ector deteried s follows: Positio w so tht its iitil poit coicide with the teril poit of The ector w is the ector represeted y the rrow fro the iitil poit of to the teril poit of w () The egtie of deoted y is defied to e the ector represeted y the rrow fro the teril poit of to the iitil poit of () The differece of w d is defied s ( w ) w (4) If k is y sclr the the sclr ltiple k of y k is defied to e the ector whose legth is k ties the legth of d whose directio is the se

33 s tht of if k > d opposite to tht of if k < We defie k d 4 Vectors i Coordite Syste Sppose we he rectglr coordite syste i -spce (or -spce) Defiitio 4 If the iitil poit of ector is the origi d the teril poit hs coordite ( ) The we write ( ) Rerk 4 (Opertios represeted i ters of coordites) If ( ) d w ( ) w w the () w ( w ) () k ( ) k k w () w ( w ) w Siilr reslts hold for ectors i -spce 4 Nor of ector ector rithetic

34 Theore 4 et d w e ectors i -spce or -spce d k d l e sclrs The () ; w w ; () ( ) ( ) (c) ; ; (d) ( ) (e) k ( l ) ( kl) ; (f) k ( ) k k ; (g) ( k l ) k l (h) ; Defiitio 4 The legth of ector is clled the or of d is deoted y Propositio 4 The R et ( ) et ( ) R The 44 Dot Prodct Defiitio 44 et d e two ectors (either oth i -spce or -spce) d θ e the gle etwee the ( θ π) The dot prodct (or clide ier prodct) of d is defied s cosθ if or ple 44 et i ( ) j ( ) d k ( ) The

35 i j i k j k The followig theore gies sipler wy to elte dot prodct Theore 44 (Copoet Forl for Dot Prodct) et ( ) ( ) R The et ( ) ( ) R Proof et d e ectors i The PQ d The R et P d Q e the teril poits of d respectiely PQ cosθ Bt PQ ( ) so PQ ( ) ( ) ( ) Fro ll these we oti ple 444 cosθ PQ () Deterie whether ( 59 ) is orthogol to ( ) () Show tht i the -spce the ector ( 6) (I geerl ( ) () Fid poit ( ) d ( ) 6y is perpediclr to the lie is perpediclr to y c ) P o the lie 5y 8 sch tht the lie throgh P Q is perpediclr to 5y 8 Theore 445 (Properties of Dot Prodct) et d w e ectors d k e sclr The

36 () () ( w) w (c) k( ) ( k) ( k) (d) iplies ple 446 () Proe: If is orthogol to oth d w the is orthogol to 8w () Proe: If d the is orthogol to Defiitio 447 (Orthogol Projectio) et d e two o-zero ectors Sppose tht c e decoposed s w sch tht is prllel to d w is perpediclr to The the ector is clled the orthogol projectio of o d is deoted y proj Theore 448 If d re two o-zero ectors the Proof et The proj proj d w The k for soe sclr k d w so k d ( w) ( w) k k w k proj

37 ple 449 () et ( c ) R d i j d k e the ectors defied i the lst sectio The proj i i ( ) proj j j ( ) d proj ( c ) () Fid the distce etwee the poit ( ) Soltio y y k P d the lie y c et Q ( ) e y poit o the lie Pt the ector ( ) t Q The D eqls the legth of Bt proj QP tht is QP D proj QP so tht its iitil poit is so ( ) ( y y ) y ( y ) y c QP D y c 45 Cross Prodct Gie two ectors d the dot prodct is sclr I this sectio we costrct ector fro d sch tht is perpediclr to oth d Defiitio 45 et ( ) ( ) R the ector i or R defied s The cross prodct of d deoted y is ( )

38 ple 45 et ( ) d ( ) () Fid () Fid ( ) ple 45 et i ( ) j ( ) d ( ) k Copte i j i k d j k ple 454 (Deterit Forl) et The i j d k e the ectors i the oe eple i j i k j k Theore 455 (Bsic Properties of Cross Prodct) et d w e ectors i R the () ( ) ( is orthogol to ) () ( ) ( is orthogol to ) (c) ( ) (grge s idetity) (d) ( w) ( w) ( )w (e) ( ) w ( w) ( w) Theore 456 (More Properties) et d w e ectors i (f) (g) (h) ( ) w ( w) ( w) R d k e y sclr the

39 (i) k ( ) ( k) ( k) (j) Theore 457 (Geoetric Iterprettio of Cross Prodct) et d e ectors i d R The is the re of the prllelogr deteried y ple 458 Fid the re of the trigle deteried y the three poits P ( ) Q ( ) d ( ) R Defiitio 459 If d w re ectors i R the the sclr ( w) is clled the sclr triple prodct of d w Rerk 45 et ( ) ( ) d w ( w w w ) The ( w) w w w w w w w w w Theore 45 () The solte le of the deterit

40 is eql to the re of the prllelogr i -spce deteried y the ectors ( ) d ( ) () The solte le of the deterit w w is eql to the ole of the prllelepiped i -spce deteried y the ectors w w w w ( ) ( ) d ( ) w ple 45 Deterie whether the three ectors ( ) ( ) d w ( ) ple Soltio re i the se The three poits re i the se ple if d oly if the ole of the prllelepiped deteried y ( ) ( ) d w ( ) is zero ie if d oly if w Now ( ) ( w ) 5 So the three gie poits re ot i the se ple 46 ies d Ples i -Spce Rerk 46 (Poit-or For) et α e ple i -spce e poit i α d ( c ) The poit P ( y z ) PQ if d oly if e ector perpediclr to α is i α if d oly if is perpediclr to PQ if d oly if ( ) ( y y ) c ( z z )

41 This is clled the poit-orl for of ple eqtio The ector is clled the or of α α Rerk 46 (Pretric qtio of ies) et l e lie i -spce throgh the poit P ( y z ) d is prllel to ( c) poit P ( y z) is o the lie l if d oly if P P is prllel to if d oly if there is sclr t sch tht P P t or y y z z This is clled the pretric eqtio of lie l t t tc Theore 46 The grph of the eqtio y cz d is ple with ( c) s orl Theore 464 The distce etwee the poit ( y z ) P d the ple y cz d is D y cz c d P ( y z )

42 Chpter 5 clide Spce R 5 clide Spce R Defiitio 5 et e positie iteger et {( ) : } R R i e the set of ll -tples of rel ers R is clled the -spce eleet ( ) is clled ector d i is clled ith copoet of the ector ple 5 we he R {( : ) R } R {( ) : R } R {( ) : R } For i i Rerk 5 eleet ( ) i write ( ) whe we tret it s ector ( ) R ector R c e regrded either s geerlized poit or ector We is clled the zero Opertios o R For R d R we he defied dditio differece d sclr ltiplictio Siilr opertios c e itrodced o y R Defiitio 54 et ( ) ( ) e ectors i S: ( ) Sclr ltiplictio: k ( k k k ) Negtie: ( ) Differece: ( ) ( ) R d k sclr we defie

43 ple 55 () ( 6 ) ( 74) ( 6) ( 7 4) ( 55 4) 4 () et ( 4) ( 48 ) R fid 4 w R sch tht w ple 56 et e ( ) e ( ) ( ) e 4 ( ) 4 ( ) R we he 4 e e e 4e 4 e Show tht for y Propositio 57 (Bsic Properties) et w e ectors i () R d () ( w) ( ) w (c) (d) ( ) (e) k ( l) ( kl) (f) k ( ) k k (g) ( k l) k l (h) k l e rel ers the: Ier Prodct Defiitio 58 et ( ) d ( ) is defied y e two ectors i K R The the ier prodct of Rerk 59 The set R with the oe opertios is clled the clide -spce ple 5 4 () If ( ) ( 7) R

44 the ( )( ) ( )( 7) ( )( ) ( )( ) 7 6 () e ie j if i j d it is if i j Theore 5 (Properties of Ier Prodct) If w e ectors i () R d () ( ) w w w (c) ( k) k( ) k l re sclrs the (d) d iff ple 5 () ( ) ( 4 ) ( ) ( 4 ) ( ) ( 4 ) ( ) ( 4) ( ) ( ) ( ) ( 4) ( ) ( ) ( )( ) ( )( ) ( 8)( ) ( )( ) ( ) ( ) ( ) () ( ) ( ) Defiitio 5 K R is defied s The or of ( ) ( ) Qestio: If d re o-zero ectors i s R c we defie the gle θ etwee d θ cos? Theore 54 (Cchy-Schwrz Ieqlity) If ( ) d ( ) K re ectors i K R the ple 55 If K re o-zero rel ers proe ( )( )

45 Orthogolity I spces R d R two ectors re orthogol to ech other if their dot prodct is zero Usig this we c defie orthogolity i y R sig ier prodct This otio hs ee fod to e ery sefl Defiitio 56 Two ectors i R re clled orthogol if ple 57 I 4 R ( ) d ( ) ( )( ) ( )( ) ( )( ) ( )( ) re orthogol sice Theore 58 (Pythgors Theore) If d re orthogol ectors i R the ple 59 Proe tht if the is orthogol to Proof iplies (( ) ( ) ) (( ) ( ) ) d so ( ) ( ) ( ) ( ) Hece ( ) ( ) ( ) ( ) ( ) ( ) iplies tht 4 ( ) So d re orthogol this 5 ier Trsfortios fro R to R Defiitio 5 (ier Trsfortio) fctio f : R R is clled lier trsfortio if for ech ( ) R f ( ) ( w w w ) R where w i s re gie y the followig lier eqtios

46 w w w M where ij re fied sclrs deteried y f ple 5 () f : R R f ( y) ( y y) is lier trsfortio fro Siilrly T : R R T ( ) ( 4 ) trsfortio () g : R g( y) ( y ) R is ot lier trsfortio R to R is lier Gie lier trsfortio ( ) ( w ) T : R y the defiitio there re sclrs ij R T w where w i i i with etries ij ij et ( ) the we see tht f ( ) for y ( ) sch tht e the tri The tri deteried y T is clled the stdrd tri of the lier trsfortio T Oiosly y lier trsfortio is iqely deteried y its stdrd tri ple 5 The stdrd tri for the lier trsfortio f i eple () is The stdrd tri for T i eple () is 4 ple 54 et T : R R e lier trsfortio whose stdrd tri is

47 Fid T ( ) ple 55 () For y lier trsfortio T the we write T T We lso se [ ] : if is the stdrd tri of T R R T to deote the stdrd tri of T () For y tri there is lier trsfortio y ( ) T for ll R etwee lier trsfortio fro T : R R defied Hece there is oe-to-oe correspodece R to R d trices ple 56 () Idetity trsfortio: If I is the idetity T I is the idetity p o () Reflectio opertors: ot the y -is et ( ) w R tri the ( ) I T I So T seds ech ector ito its syetric ige : R R T The w w y So T is lier d the stdrd tri of T is () Rottio: et T : R R e the p tht rottes ech ector throgh fied gle θ The for ech ( y) T ( cosθ y siθ siθ y cosθ ) the stdrd tri for this rottio is cosθ siθ siθ cosθ So Copositio of ier Trsfortio Qestio: Is the copositio T o T of two lier trsfortio still lier trsfortio? T T? If yes wht is the reltio etwee [ ] T T o d [ ] [ ] Theore 57 If T s s : R R T : R R re oth lier trsfortios the T o T T T lier d [ ] [ ][ ] T o T : R R is ple 58

48 T R T ( y ) ( y y ) : R T R : R The ( T T )( y ) T ( y y ) ( y y ) o Hece [ T ot ] [ T ][ T ] 5 Properties of ier Trsfortios Oe-to-oe ier Trsfortios ple 5 The orthogol projectio : R R 4 f 5 f defied y ( y ) ( ) fctio For eple f ( ) ( ) ( ) t ( 4 ) ( 5 ) f is ot oe-to-oe The lier trsfortio T : R R tht rottes ech ector gle θ is oe-to-oe It seds differet ectors to differet ectors Notice tht [ f ] is ot iertile d [ ] cosθ siθ f siθ cosθ [ ] [ T ] T is iertile So whether trsfortio is oe-to-oe is closely relted to the iertiility of its stdrd tri Defiitio 5 lier trsfortio T : R R is sid to e oe-to-oe if it ps distict ectors i R to distict ectors i R Theore 5 If is tri the the followig stteets re eqilet: () T is oe-to-oe () is iertile (c) The rge of T is R where the rge of { R } T is T ( ) : Rerk 54 This theore oly pplies to lier trsfortio fro trsfortios R to itself ot to ritrry lier

49 ple 55 If 4 the T : R R is ot oe-to-oe ecse is ot iertile Defiitio 56 The ierse f of oe-to-oe p f : C D is p f : D C ( f o f )( ) for ll C d ( f o f )( y) y for ll y D stisfyig Propositio 57 If T : R R is the ltiplictio y d T is oe-to-oe the the ierse of T eists d eqls to T ple 58 Show tht the lier trsfortio T : R R defied y the eqtios is oe-to-oe d fid ( w w ) Soltio The stdrd tri of T is So T T T T ( w w ) w w w w Hece ( w w ) w w w w 4 T w w 5 5 w w 5 5

50 Chrcteristic of ier Trsfortio Theore 59 p T : R R is lier trsfortio if d oly if it stisfies the followig two coditios: T T T for ll ectors () ( ) ( ) ( ) () T ( c ) ct ( ) i R d y sclr c Rerk 5 If T : R R is lier the for y ectors t d sclrs k k k t T ( k k k ) k T ( ) k T ( ) k T ( ) t t t t Defiitio 5 et e ( ) e ( ) e ( ) { e e } sis of R e is clled the stdrd Theore 5 If T R R the stdrd tri for T is : is lier trsfortio d { e e } e [ T ] T ( e ) T ( e ) T ( e ) is the stdrd sis of [ ] R the ple 5 et T : R R e defied y T ( y ) ( y y ) The T ( e ) T ( ) ( ) T ( e ) T ( ) ( ) T [ e ] [ ] T ( e ) T ( ) so the stdrd tri for T is 54 igeles d igeectors of ier Opertor Defiitio 54 et T : R R e lier opertor sclr λ is clled eigele of T if there is o-zero ector i R sch tht T () λ

51 Rerk 54 () If T : R R is the ltiplictio y the T ( ) λ if d oly if λ So λ is eigele of T if d oly if it is eigele of () If λ is eigele of d is oe eigeector correspodig to λ The ( λ I ) s so det ( λ I ) Tht is ech eigele of stisfies this eqtio Coersely if λ stisfies this eqtio the it is eigele of ple 54 T e the lier trsfortio defied y T ( y) ( y y) et : R R stdrd tri of T is So λ is the oly eigele of T λ det λ ( λi ) ( λ )( λ ) The