Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1

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1 Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida sac R is simly th st of all ordrd -tuls of ral umbrs x = ( x 1, K, x ) Thus R 1 is simly th ral umbrs, R is th la, ad R 3 is Euclida thr-sac Ths ordrd -tuls ar calld oits, or vctors This dfiitio dos ot cotradict our rvious dfiitio of a vctor i cas =3 i that w idtifid ach vctor with a ordrd tril ( x1, x, x3 ) ad sok of th tril as big a vctor W ow dfi various arithmtic oratios o R i th obvious way If w hav vctors x = ( x 1, K, x ) ad y = ( y 1, y, K, y ) i R, th sum x + y is dfid by x + y = ( x + y, x + y, K 1 1, x + y ), ad multilicatio of th vctor x by a scalar a is dfid by ax = ( ax, ax, K 1, ax ) It is asy to vrify that a( x + y) = ax + ay ad ( a + b) x = ax + bx Agai w s that ths dfiitios ar tirly cosistt with what w hav do i dimsios 1,, ad 3-thr is othig to ular Cotiuig, w dfi th lgth, or orm of a vctor x i th obvious mar x = x 1 + x + K + x Th scalar roduct of x ad y is 51

2 x y = x y + x y + K + x y = x y 1 1 i i i = 1 It is agai asy to vrify th ic rortis: x = x x 0, ax = a x, x y = y x, x ( y + z) = x y + x z, ad ( ax) y = a( x y) Th gomtric laguag of th thr dimsioal sttig is rtaid i highr dimsios; thus w sak of th lgth of a -tul of umbrs I fact, w also sak of d( x, y) = x y as th distac btw x ad y W ca, of cours, o logr rly o our vast kowldg of Euclida gomtry i our rasoig about R wh > 3 Thus for 3, th fact that x + y x + y for ay vctors x ad y was a siml cosquc of th fact that th sum of th lgths of two sids of a triagl is at last as big as th lgth of th third sid This iquality rmais tru i highr dimsios, ad, i fact, is calld th triagl iquality, but rquirs a sstially algbraic roof Lt s s if w ca rov it Lt x = ( x 1, K, x ) ad y = ( y 1, y, K, y ) Th if a is a scalar, w hav Thus, ax + y = ( ax + y) ( ax + y) 0 ( ax + y) ( ax + y) = a x x + ax y + y y 0 This is a quadratic fuctio i a ad is vr gativ; it must thrfor b tru that 4( x y) 4( x x)( y y) 0, or x y x y This last iquality is th clbratd Cauchy-Schwarz-Buiakowsky iquality It is xactly th igrdit w d to rov th triagl iquality 5

3 x + y = ( x + y) ( x + y) = x x + x y + y y Alyig th C-S-B iquality, w hav x + y x + x y + y = ( x + y ), or x + y x + y Corrsodig to th coordiat vctors i, j, ad k,,, K, ar dfid i R by 1 M 1 3 = ( 10,, 00,, K, 0) = ( 0100,,,, K, 0) = ( 0010,,,, K, 0), = ( 000,,, K, 01, ) th coordiat vctors Thus ach vctor x = ( x 1, K, x ) may b writt i trms of ths coordiat vctors: x = x ii i= 1 Exrciss 1 Lt x ad y b two vctors i R Prov that x + y = x + y x y if ad oly if = 0 (Adotig mor gomtric laguag from thr sac, w say that x ad y ar rdicular or orthogoal if x y = 0) Lt x ad y b two vctors i R Prov a) x + y x y = 4 x y b) x + y + x y = [ x + y ] 53

4 3 Lt x ad y b two vctors i R Prov that x y x + y 4 Lt P R 4 b th st of all vctors x = ( x1, x, x3, x4 ) such that 3x + 5x x + x = Fid vctors ad a such that P = { x R 4 : ( x a) = 0 } 5 Lt ad a b vctors i R, ad lt P = { x R : ( x a) = 0 a)fid a quatio i x, x, K, ad x such that x = ( x, x, K, x ) P if ad oly if 1 1 th coordiats of x satisfy th quatio b)dscrib th st P b i cas = 3 Dscrib it i cas = [Th st P is calld a hyrla through a] 5 Fuctios W ow cosidr fuctios F: R R Not that wh = = 1, w hav th usual grammar school calculus fuctios, ad wh = 1 ad = or 3, w hav th vctor valud fuctios of th rvious chatr Not also that xct for vry scial circumstacs, grahs of fuctios will ot lay a big rol i our udrstadig Th st of oits ( x, F ( x)) rsids i R + sic x R ad F ( x) R ; this is difficult to s ulss + 3 fuctio F: R W bgi with a vry scial kid of fuctios, th so-calld liar fuctios A R is said to b a liar fuctio if i) F ( x + y) = F( x) + F( y) for all x, y R, ad ii)f ( ax) = af ( x) for all scalars a ad x R Examl Lt = = 1, ad dfi F by F ( x) = 3 x Th F ( x + y) = 3( x + y) = 3x + 3y = F ( x) + F ( y) ad F ( ax) = 3( ax) = a3 x = af ( x) This F is a liar fuctio 54

5 Aothr Examl Lt F: R R 3 b dfid by F( t) = ti + tj 7tk = ( t, t, 7 t ) Th F( t + s) = ( t + s) i + ( t + s) j 7( t + s) k = [ ti + tj 7tk] + [ si + sj 7sk] = F ( t) + F( s) Also, F( at ) = ati + atj 7at k = a[ ti + tj 7tk] = af( t ) W s yt aothr liar fuctio O Mor Examl Lt F: R R b dfid by 3 4 F (( x, x, x )) = ( x x + 3x, x + 4x 5x, x + x + x, x + x ) It is asy to vrify that F is idd a liar fuctio A traslatio is a fuctio T:R R such that T( x) = a + x, whr a is a fixd vctor i R A fuctio that is th comositio of a liar fuctio followd by a traslatio is calld a affi fuctio A affi fuctio F thus has th form F ( x) = a + L( x), whr L is a liar fuctio Examl Lt F: R R 3 b dfid by F( t) = ( + t, 4t 3, t) Th F is affi Lt a = (, 4, 0 ) ad L( t) = ( t, 4 t, t ) Clarly F ( t) = a + L( t) Exrciss 6 Which of th followig fuctios ar liar? Exlai your aswrs a) f ( x) = 7 x b) g( x) = x 5 c) F ( x1, x) = ( x1 + x, x1 x, 3x1, 5x1 x, x 1 ) d) G( x, x, x ) = x x + x ) F ( t) = ( t, t, 0, t ) f) h( x1, x, x3, x4) = ( 1, 0, 0) g) f ( x) = si x 55

6 7 a)dscrib th grah of a liar fuctio from R to R b)dscrib th grah of a affi fuctio from R to R 56