MODUL III RUANG VEKTOR

Transkripsi

1 MODUL III RUANG VEKTOR.. Rug Vetor Rug etor merup mteri yg sgt petig dlm Mtemti d Sttisti. Utu memgu rug etor diperlu pegethu tetg sistem ilg seperti ilg rel tu ilg Komples esert opersi pejumlh d perli dri ilg terseut. Wlupu my rug etor tid errti oye-oye dri rug terseut erup etor dlm rti yg seery tetpi oye terseut dpt erper segi etor sl memeuhi sift dri rug etor. Beriut dieri defiisi rug etor ts ilg rel R. Defiisi. ( Rug Vetor ) Dieri rug V yg dilegpi deg opersi pejumlh (+) d perli (.) deg slr ts ilg rel R. Seljuty misl u w V d merup slr-slr dlm R. Rug V diseut rug etor ts ilg rel R ji memeuhi : A. Terhdp opersi pejumlh ersift : A. u + V. ( Tertutup terhdp pejumlh ) A. u + = + u. ( Komuttif terhdp pejumlh ) A. ( u + ) + w = u + ( + w ). ( Assositif deg pejumlh ) A4. Utu setip u V terdpt V sehigg : + u = u +. ( Ad eleme etrl ) A5. Utu setip u V terdpt -u V sehigg : u + (-u) = (-u) + u =. ( Ad iest ) B. Terhdp opersi Perli deg slr ersift : B. u V. ( Tertutup terhdp perli deg slr) B. (u + ) = u +. B. ( + ) u = u + u. B4. (u) = ()u. B5. u = u. Beriut ii dieri cotoh-cotoh rug etor ts ilg rel R.

2 Cotoh. Dieri etor-etor u R = RxR = { () ; R d R }. Pejumlh d perli deg slr didefiisi segi eriut : u + = ( ) + (c d) = (+c +d) d u = ( ) R. Perliht hw R deg opersi di ts merup rug etor ts ilg rel R. Jw : Utu memperliht R merup rug etor diselidii semu sift A A5 d B B5. Amil semrg u w R d slr R m u w dpt disji mejdi : u = ( ) = (c d) d w = (e f) deg c d e f R. A. Terhdp opersi pejumlh ersift : A. u + = (+c +d) R. Se c d R m + c R d + d R. A. u + = (+c +d) = (c+ d+). ( pejumlh ilg rel omuttif ) = (c d) + ( ) = + u. A. ( u + ) + w = (+c +d) + (e f) = ((+c)+e (+d)+f) = (+(c+e)+(d+f)). (pejumlh ssositif) = () + (c+e d+f) = u + ( + w ). A4. Utu setip u R terdpt = () R sehigg : + u = () + ( ) = () = u u + = ( ) + () = () = u A5. Utu setip u R terdpt -u = (- -) R sehigg : u + (-u) = () + (--) = (+(-) +(-)) = ( ) 44

3 = R. (-u) + u = (- -) + ( ) = (-+ -+) = ( ) = R. B. Terhdp opersi Perli deg slr ersift : B. u = ( ) = ( ) R. Se : R d R m R d R. B. (u + ) = (+c +d) = (+c +d) = ( ) + ( c d) = () + (cd) = u +. B. ( + ) u = ( + ) ( ) = (( + ) ( + )) = ( + + ) = ( ) + ( ) = ( ) + ( ) = u + u. B4. (u) = ( ) = (() ()) = (() ())= ()( ) = ()u. B5. u = () = ( ) = () = u. Kre syrt A A5 d B B5 dipeuhi m rug R merup rug etor ts ilg rel R. Cotoh. Dieri etor u R = { ( ) ; j R j= }. Didefiisi pejumlh d perli deg slr segi eriut : u + = ( ) + ( ) = ( ) j R j R j= d u = ( ) R. 45

4 Perliht hw R deg opersi-opersi di ts merup rug etor ts ilg rel R. Jw : Sol ii merup geerlissi dri Cotoh. sehigg peyelesiy serup deg cotoh terseut. Co d selesi segi ltih. Cotoh. Dieri rug R(f) yg meyt himpu semu fugsi f pd gris rel R. Utu setip f g R(f) d slr R opersi pejumlh d perli deg slr megiuti : f+g = (f+g)( = f(+g( d f = (f)( = f( x R. Aph R(f) merup rug etor ts ilg rel R?. Jw : Utu memperliht R(f) merup rug etor hruslh memeuhi semu sift A-A5 d B-B5. Amil semrg fgh R(f) d slr R m f g h dpt diyt mejdi : f = f( R g = g( R d h = h( R x R. A. Terhdp opersi pejumlh ersift : A. f + g = (f+g)( = f(+ g( R(f). Se f( R d g( R m f( + g( R. B. f + g = (f+g)( = f(+ g( = g(+ f( ( pejumlh ilg rel omuttif ) = (g+f)( = g + f. B. (f + g) + h = ((f+g) + h )( = (f+g)( + h( = (f(+g() + h( = f(+ (g(+ h() (pejumlh ilg rel ssositif) = (f + (g+h))( = f + (g + h) A4. Utu setip f R(f) terdpt = ( R(f) sehigg : 46

5 + f = (+f)( = (+ f( = f( = f. f + = (f+)( = f(+ ( = f( = f. A5. Utu setip f R(f) terdpt -f = -f( R(f) sehigg : f + (-f) = (f +(-f))( = f( + -f( = ( R(f). (-f) + f = ((-f) + f)( = -f( + f( = ( R(f). B. Terhdp opersi Perli deg slr ersift : B. f = (f)( = f( R(f). Se f( R m f( R B. (f + g ) = ((f+g))( = (f+g)( = (f( + g() = f( + g( = (f)( + (g)( = (f + g)( = f + g. B. ( + ) f = (( + ) f)( = ( + )f( = f( + f( = (f)( + (f)( = (f + f)( = f + f. B4. (f) = ((f))( = ((f)() = (f() = ()f( = ()f. B5. u = (f)( = f( = f( = f. 47

6 Kre syrt A A5 d B B5 dipeuhi m rug R(f) merup rug etor ts ilg rel R. Cotoh.4 Dieri rug M x (D) deg : M x (D) = { Mtris eruur x eretu deg R}. Opersi pejumlh d perli deg slr megiuti opersi pejumlh d perli deg slr pd mtris. Aph M x (D) merup rug etor ts ilg rel R? Jw : Utu memperliht M x (D) merup rug etor diselidii semu sift dri rug etor. Amil semrg u w M x (D) d slr R m u w dpt ditulis mejdi : u = c e = d w = d deg c d e f R. f A. Terhdp opersi pejumlh : c c A. u + = + = d M x (D). d Se c d R m + c R d + d R. c A. u + = d c = ( pejumlh ilg rel omuttif ) d c = + d = + u. c e A. ( u + ) + w = + d f ( c) e = ( d) f 48

7 49 = ) ( ) ( f d e c (pejumlh ssositif) = + f d e c = u + ( + w ). A4. Utu setip u M x (D) terdpt = M x (D) sehigg + u = + = = u. u + = + = = u. A5. Utu u M x (D) terdpt -u = Mx (D) sehigg : u + (-u) = + = ) ( ) ( = = M x (D). (-u) + u = + = ) ( ) ( = = M x (D). B. Terhdp opersi perli deg slr ersift : B. u = = Mx (D). Se : R d R m R d R.

8 5. (u + ) = d c = ) ( ) ( d c = d c = + d c = + d c = u +. B. ( + ) u = ( + ) = ) ( ) ( = = + = + = u + u. B4. (u) = = ) ( ) ( = ) ( ) ( = ()

9 = ()u.. B5. u = = =. = u. Kre syrt A A5 d B B5 dipeuhi m rug : M x (D) = { Mtris eruur x eretu deg R } Deg opersi pejumlh d perli deg slr yg erlu pd mtris merup rug etor ts ilg rel R. Ji d perhti Cotoh. smpi deg Cotoh.4 d meliht hw rug-rug etor terseut semuy merup rug etor ts ilg rel R. Pd dsry rug etor tid sellu ts ilg rel R tetpi d jug rug etor ts ilg Komples C. Nmu tid disji pd modul ii. Mugi d erty hw ph semu rug deg opersi pejumlh d perli deg slr merup rug etor?. Jwy dlh tid. Apil rug yg dieri deg opersi pejumlh d perli deg slr tid memeuhi slh stu dri sift A A5 tu B B5 m rug terseut ulh rug etor. Beriut dieri eerp cotoh rug yg u rug etor ts ilg rel R. Cotoh.5 Dieri rug R + deg : R + = { (xy) R ; x y x R y R } yitu himpu semu psg erurut (xy) yg terlet pd udr pertm. Seljuty opersi pejumlh d perli deg slr megiuti opersi pejumlh d perli pd rug R. Aph R + rug etor ts ilg rel R?. Jw : Rug R + u rug etor ts ilg rel R re syrt B- tid dipeuhi. Amil semrg u R + m u dpt diyt mejdi : u = (xy) R deg x R y R d x y. Dieri slr - R > m : -u = (-x -y) R + 5

10 se -x d -y utu x y. Cotoh.6 Dieri rug M x (D*) deg : M x (D*) = { Mtris eruur x eretu deg R}. Opersi pejumlh d perli deg slr megiuti opersi pejumlh d perli deg slr yg d pd mtris. Aph M x (D*) merup rug etor ts ilg rel R? Jw : Rug M x (D*) u rug etor ts ilg rel R re syrt A tid dipeuhi. Amil semrg u M x (D*) m u dpt diyt mejdi : u = u + = c = deg c d R. d c c + = d M x (D*). c d Dlm y persol prtis yg meygut rug etor serig diperhti rug-rug li yg merup gi dri rug etor terseut yitu surug (rug gi). Himpu W yg merup himpu gi dri rug etor V dit surug V ji W dlh rug etor ts opersi pejumlh d perli deg slr yg didefiisi pd rug etor V. Ad telh megethui deg i hw utu memperliht sutu rug merup rug etor hruslh memeuhi syrt A-A5 d B-B5. Ji d iuti prosedur terseut secr detil stu perstu m melit peerj yg cuup pjg d urg prtis. Beriut dieri sutu teorem utu memperliht rug W merup surug dri rug etor V ts ilg rel R. Teorem. ( Surug ) Ji V rug etor d W himpu gi dri V m rug W merup surug dri V ji erlu : (i). (u W) ( u+ W ). ( tertutup terhdp pejumlh ). 5

11 (ii). ( R u W) (u W ). (tertutup terhdp perli slr). Cotoh.7 Dieri rug R(f[]) deg : R(f[]) = { f ; f fugsi erili rel pd iterl [] R }. Opersi pejumlh d perli deg slr dieri oleh : f + g = (f+g)( = f( + g( d f = (f)( = f( x R. Seljuty dieri rug : C[] = { f ; f fugsi otiu pd iterl [] R }. Ap rug C[] merup surug dri rug etor R(f[]).? Jw : Kit telh meuju R(f[]) deg opersi-opersi yg dieri merup rug etor ts ilg rel (liht Cotoh.). Rug C[] merup surug dri R(f[]) se : (i). Amil semrg fg C[] m : f = f( d g = g( fugsi-fugsi otiu pd iterl []. Aity : f+g = (f+g)( = f( + g( C[]. Se pejumlh du fugsi yg msig-msig otiu dlh otiu. (ii). Utu R diperoleh : f = (f)( = f( C[]. Se perli fugsi otiu deg slr ilg rel dlh fugsi otiu. Cotoh.8 Dieri rug-rug : M( = { Mtris eruur x } d M() = { Mtris eruur x eretu R }. Opersi pejumlh d perli deg slr sesui deg opersi yg erlu pd mtris. Perliht M() merup surug dri M(. 5

12 Jw : Amil semrg AB M() d slr R m : c A = B = d A = d cd R. c c (i). A + B = + = d M(). d Se cd R m +c R d +d R. (ii). A = M() Se R d R m R d R. Kre (i) d (ii) erlu m M() merup surug dri rug etor M(. Cotoh.9 Dieri rug etor : R(f[]) = { f ; f fugsi erili rel pd iterl [] R }. Seljuty dieri sutu rug : [ ] = { f ; f poliomil derjt pd iterl [] R }. Perliht [ ] merup surug dri rug etor R(f[]). Jw : Amil semrg fg [ ] d slr R m f d g dpt diyt mejdi : f = f( = g = g( = x x... x d x x... x deg... R d... R. (i). f + g = (f+g)( = f( + g( = ( x x... x ) + ( x x... x ) = ( ) x ) ( ) x ( ) x... ( [ ]. Se... R d... R m 54

13 ( ) R... ( ) R d m. Jdi f+g merup poliomil derjt. (ii). f = (f)( = f( = ( x x... x ) = ( ) ( ) x ( ) x... ( ) x [ ]. Se d... R m : R... R d. Jdi f merup poliomil derjt. Kre (i) d (ii) erlu m [ ] merup surug dri rug etor R(f[])... Bsis d Dimesi Rug Vetor Pd pemhs seelumy d telh memhmi d dpt memperliht sutu rug merup rug etor deg meggu defiisi rug etor. Pd gi ii d dihrp mmpu memperoleh etor-etor (deg sift tertetu) yg memgu d merup erg dri rug etor terseut esert dimesiy. Sift tertetu yg dimsud disii dlh etoretor yg es lier (idepedet lier). Sutu etor-etor yg es lier d memgu/meretg/sp rug etor V diseut sis dri V. Deg demii utu mempeljri sis dri rug etor diperlu pegerti tetg osep-osep eriut :. Komisi lier dri sutu etor.. Vetor-etor yg memgu sutu rug etor. c. Vetor-etor yg es lier. Pertm dieri pegerti tetg omisi lier dri sutu etor d Vetoretor yg memgu/meretg/sp sutu rug etor. Defiisi. (Komisi Lier) Dieri etor-etor ji dpt diyt segi :. Vetor dit omisi lier dri 55

14 =... = i deg i i =... merup slr ilg rel. i i Defiisi. (Memgu) Dieri etor-etor... pd rug etor V. Ji etor-etor pd V dpt diyt segi omisi lier dri... m etor-etor... dit memgu/ meretg/sp dri rug etor V. Cotoh. Dieri etor-etor dlm R : = (-) = (64) d = (97) * = (4-8). Perliht :. Vetor merup omisi lier dri d.. Vetor * u merup omisi lier dri d. Jw :. Vetor merup omisi lier dri d hruslh terdpt R d R sehigg : = + yitu : (97) = (-) + (64) = ( - ) + (6 4 ) = ( ) Persm ii memeri : +6 = 9 () +4 = () - + = 7. () Persm () d () memeri : 8 = 6 tu =. Dri persm () deg mesustitusi = memeri : = -. 56

15 Aity etor dpt diyt segi omisi lier dri d yitu : = Vector * merup omisi lier dri d hruslh terdpt R d R sehigg : * = +. (4-8) = ( ) Persm ii memeri : +6 = 4 () +4 = - () - + = 8. () Persm () d () memeri : 8 = tu = 5. (4) Persm () digd deg emudi diurgi deg persm () diperoleh : 8 = 9 tu = 9/8. (5) Persm (4) d persm (5) memperliht hw tid d d sehigg : * = +. Jdi * u omisi lier dri d. Cotoh. Dieri himpu poliomil { x x...x } yg didefiisi pd iterl []. Aph poliomil ii memgu [ ]? [ ] = { f ; f( = Jw : x x... x pd [] }. Poliomil { x x...x } memgu [ ]. Se utu semrg f [ ] dpt diyt mejdi : f( = x x... x deg... R. Cotoh. 57

16 Dieri etor-etor dlm R R d R : = () d = () w = () w = () d w = (). x = (...) x = (...)... d x = (...). Aph :. { = () d = () }. { w = () w = () d w = () } c. { x = (...) x = (...)...x = (...) } msig-msig memgu R R d R. Jw :. { = () = () } memgu R. Utu semrg etor R dpt diyt segi omisi lier dri : = () = () + () = +.. { w = () w = () d w = ()} memgu R. Utu semrg etor w R dpt diyt segi omisi lier dri : w = (c) = () + () + c () = w + w + c w c. { x = (...) x = (...)...x = (...) } memgu R. Utu semrg etor u R dpt diyt segi omisi lier dri : u = ( ) = ( ) + ( ) + + ( ) = x + x + + x. Setelh d memhmi osep omisi lier d memgu eriut ii disji defiisi etor-etor yg es lier d sgt ergu dlm memperoleh sis dri sutu rug etor. Defiisi.4 ( Vetor Bes Lier) Dieri himpu etor {... } dlm rug etor V. 58

17 (i). Vetor {... } diseut es lier (idepedet lier) ji omisi lier :... = mempuyi peyelesi = = = =. (ii). Vetor {... } dit tid es lier (depedet lier) ji omisi lier :... = mempuyi peyelesi plig tid d stu j j =. Defiisi.5 ( Bsis ) Misl V rug etor d B = {... } merup himpu erhigg dri etor-etor pd rug etor V. B diseut sis utu rug etor V ji : (i). B = {... } es lier d (ii). B = {... } memgu V. Cotoh. Dieri etor-etor dlm R R d R : = () d = () R w = () w = () d w = () R. x = (...) x = (...)... d x = (...) R. Aph :. B = { } sis utu R.. B = { w w w } sis utu R. c. B ={ x x...x } sis utu R. Jw : Utu memperliht B B B sis utu rug R R d R dituju :.(i). B = { } es lier se omisi lier : = () + () = 59

18 ( ) = (). mempuyi peyelesi = =. Jdi B = { } es lier. (ii). B = { } memgu rug R (liht cotoh.). Kre (i) d (ii) m B = { } merup sis(sis u) utu rug etor R..(i). B = { w w w } es lier se omisi lier : w + w + w = () + () + () = ( ) = (). mempuyi peyelesi = = =. Jdi B = { w w w } es lier. (ii). B = {w w w } memgu rug R (liht cotoh.). Kre (i) d (ii) m B = { w w w } merup sis(sis u) utu rug etor R. c.(i). B ={ x x...x } es lier se omisi lier : x + x + + x = ( ) + ( ) + + ( ) = ( ) = ( ). mempuyi peyelesi = = = =. Jdi B ={ x x...x } es lier. (ii). B ={x x...x } memgu rug R (liht cotoh.). Kre (i) d (ii) m B ={ x x...x } merup sis(sis u) utu rug etor R. Perlu d ethui hw sis dri sutu rug etor tidlh tuggl (tid stu-stuy). Beriut dieri sutu cotoh yg memer peryt ii. Cotoh.4 Dieri etor-etor pd rug etor R : B = { = () = () d = (c) } B = { w = () w = () d w = () } 6

19 B = { u = () u = (9) d u = (4) }. Buti B sis utu rug etor R.. Buti B sis utu rug etor R. c. Buti B sis utu rug etor R. Jw : Utu memperliht B B B sis utu rug R dituju.(i). B = { } es lier se omisi lier : + + = () () + () + (c) = () ( c) = (). = = d c =. mempuyi peyelesi = = =. Jdi B = { } es lier. (ii). B = { } memgu rug R. Se utu semrg dlm R dpt diyt segi omisi lier : = (c) = () + () + (c) = + + Kre (i) d (ii) m B = { } merup sis utu rug etor R.. B = { w = () w = () d w = () } merup sis u utu R (liht cotoh.). c. B = { u u u } jug merup sis utu R re B memgu R d es lier. Ad dpt memperliht eer peryt ii segi ltih. Cotoh.6 Dieri himpu B* = { M M M M 4 } deg : M = M = M = M 4 =. 6

20 Misl M( = { Mtris x eretu c d }. Tuju B* merup sis (sis u) utu M(. Jw : (i). B* = { M M M M 4 } es lier se : M + M + M + 4 M 4 = = = = 4 = = = 4 =. Jdi B*= { M M M M 4 } es lier. (ii). B* memgu rug etor M(. Se utu semrg M = c dpt diyt segi omisi lier : c d didlm M( d = c d = + + c + d = M + M + c M + d M 4 Kre (i) d (ii) m B* merup sis (sis u) utu rug etor M(. Ad telh megethui dri cotoh-cotoh seelumy hw sis dri sutu rug etor tid tuggl. Wlupu demii yy etor dlm sis yg ered-ed terseut dlh sm. Keer peryt ii dieri oleh teorem eriut. Teorem. Misl V sutu rug etor ts ilg rel R. 6

21 Ji B={... } d B*= { u u...u r } msig-msig merup sis utu rug etor V m = r. Utu memperliht eer teorem di ts d dpt memperhti pejels eriut. Kre B={... } d B*= { u u...u r } merup sis utu rug etor V m : (i). Vetor u u...u r es lier. Kre B sis m r. (ii). Vetor... jug es lier. Kre B* jug sis utu V m r. Kre (i) d (ii) m r =. Cotoh.7 Dieri etor-etor pd rug etor R : B = { } deg = () d = (). B = { u u } deg u = (4) d u = (4). B = { w w } deg w = () d w = ().. Aph B B B sis-sis utu rug etor R.. Kesimpul p yg d dpt peroleh dri (). Jw :. Ad dpt deg mudh memperliht hw : B ={ =() = ()} B ={ u =(4) u = (4)} B ={ w =() w =()} es lier d memgu R. Sehigg B B B sis-sis utu R. Co d perliht eer ii segi ltih.. Terliht hw B = { } B={ u u } d B={ w w} msigmsig memut sey du etor. Jdi B B B sis-sis yg erli dri rug etor R tetpi etigy mempuyi etor peyusu sis yg sm yitu du etor. Byy etor peyusu sutu sis merup hl yg sgt petig dlm meetu dimesi dri sutu rug etor. Defiisi eriut memeri pegerti tetg dimesi dri rug etor V ts ilg rel R. 6

22 Defiisi.6 ( Dimesi Rug Vetor ) (i). Ji V rug etor m dimesi V ditulis deg dim(v) dlh yy etor-etor yg meyusu sis V. (ii). Rug etor ol mempuyi dim(v) =. Cotoh.8 Dieri himpu-himpu B D d A deg :. B = {() () ()} sis utu rug etor R.. D = { x x x } sis utu rug etor [ ]. c. A = {... } sis utu rug etor V. Tetu dimesi dri rug etor R [ ] d V. Jw :. Kre B = {() () ()} sis utu rug etor R m dim(r ) =. Se B memut tig etor peyusu yitu : () () d ().. Kre D = { x x x } sis utu rug etor [ ] m dim( [ ] ) = +. Se D memut + etor yitu : x x x. c. Kre A = {... } sis utu rug etor V m dim(v)=. Se A memut etor peyusu sis. Perlu d igt hw utu memperoleh sis d dimesi sutu rug etor sgt tergtug pd sift ees lier yg memgu rug terseut. Beriut disji eerp peryt petig ph etor-etor dlm sutu rug etor es lier tu tid. (). Misl R(f) rug etor dri himpu fugsi-fugsi erili rel. Fugsifugsi fgh R(f) mempuyi turu pertm d edu (diferesiel) pd xr. Ji Wrosi : f ( f ( f ( W( = g( h( g( h( g( h( m f g h es lier. 64

23 (). Ji B = {... } sis utu rug etor V m setip himpu deg leih dri etor dlh tid es lier. Cotoh.9 Dieri himpu B = { x x } d D = { x e x }. Aph B d D msig-msig es lier? Jw : Misl B = { f( = g( = x h( = x } d D = { f( = g( = x h( = e x }. Utu memperliht B es lier diperhti Wrosi : f ( f ( f ( W( = g( g( g( h( h( h( = x =. x x Jdi B = { x x } es lier. Utu memperliht D es lier diperhti Wrosi : f ( f ( f ( W( = g( g( g( h( h( h( = e x x e x e x = e x utu setip xr. Jdi D = { x e x } es lier. Beriut ii dieri seuh osep yg sgt petig dlm Mtemti d Sttisti yitu r sutu mtris. R sutu mtris umumy dircg d didesi erdsr pegerti tetg ees lier Bsis d etor-etor peyusu dri sutu sis. Beriut dieri eerp defiisi d teorem yg erit deg r sutu mtris. Defiisi.7. (Rug Bris d Rug Kolom Mtris) 65

24 Dieri mtris A eruur mx etor ris r i i =...m d etor olom j j =.. : A = r = : m : m :... : m K = : m... K = : r m = m m... m Vetor-etor r =. m r m = m m... m yg teretu dri ris-ris mtris A diseut etor-etor ris A. Seliy etor-etor olom : K = : m... K = : m yg teretu dri olom-olom mtris A diseut etor-etor olom A. Su rug R yg digu oleh etor-etor ris dim rug ris A. Su rug R m yg digu oleh etor-etor olom dim rug olom A. Defiisi.8. ( R ) R mtris A ditulis deg simul r(a) tu r(a) dieri oleh : r(a) = dim(rug ris A) = dim(rug olom A). Teorem. ( Dimesi Rug Bris d Rug Kolom ) Ji A mtris eruur mx m : dim(rug ris A) = dim(rug olom A). Teorem.4 (Keer Rug Bris) Opersi elemeter mtris tid meruh rug ris sutu mtris. Teorem.5 Ji A mtris eruur x m peryt eriut euile : (). A mempuyi iest. (). det(a). (c). r(a) =. 66

25 67 (d). Vetor-etor ris A es lier. (e). Vetor-etor olom A es lier. Berdsr Defiisi.7 mtris A eruur m x dpt disji mejdi A = r m r r : tu A =.... Cotoh.. Dieri mtris A eruur x4 eriut : A = Sji mtris A dlm etu etor-etor ris.. Sji mtris A dlm etu etor-etor olom. c. Tetu sis d dimesi rug ris mtris A. d. Tetu sis d dimesi rug olom mtris A. e. Tetu r mtris A. Jw :. Mtris A disji mejdi etor-etor ris r r d r : A = = : m R R R deg r = [ ] r = [ 5 ] d r = [ 4 4-4].. Mtris A disji mejdi etor-etor olom d 4 : A = = 4 deg = = 4 = 4 5 d 4 = 4.

26 68 c. Utu memperoleh sis d dimesi rug ris dri mtris A digu opersi ris elemeter dri sutu mtris d memeri = u Co d perliht eer ii segi ltih. Kre d du etor ris yitu u d : u = [ ] d = [ -] yg tid sm deg ol m sis rug ris mtris A : B = {u = [ ] = [ -]}. Jdi dim(rug ris mtris A) =. d. Utu memperoleh sis d dimesi rug olom mtris A digu opersi ris elemeter mtris. Pertm mtris A dimil trsposy diperoleh : T A = Deg opersi elemeter ris utu mtris A diperoleh : = z y Co d perliht eer ii segi ltih. Kre d du etor ris yitu y d z : y = [ ] d z = [ ] yg tid sm deg ol m sis rug ris mtris T A : { y = [ ] z = [ ]}. Secr euile diperoleh sis rug olom dri mtris A dlh : K = { = T y = = T z = }. Kre d du etor olom yitu d yg tid sm deg

27 ol m dim(rug olom mtris A) =. e. r(a) = dim(rug ris A) = dim(rug olom A) =... Bsis Ortoorml Pd pemhs seelumy d telh megethui hw sis dri sutu rug etor tid tuggl rtiy sutu rug etor V dpt mempuyi sis leih dri stu. Keyt ii memeri pelug epd d utu memilih sutu sis tertetu dri rug etor. Kre Ad es memilih sutu sis dihrp diperoleh peyelesi yg leih mudh utu sutu prolem yg erit deg rug etor. Dlm persol-persol Mtemáti d Sttisti serig diperhti sis-sis yg ortogol d sis yg ortoorml. Beriut dieri sutu metode utu memgu sis yg ortoorml dlm sutu rug tetpi seelumy dieri pegerti tetg ier product (hsil li dlm) orm d rug-rug etor husus seperti rug ier product (pre-hilert/hsil li dlm) d rug erorm rug Hilert sert rug Bch. Defiisi.9 (Ier Product) Dieri rug etor V ts ilg rel R.. Ier product < u > pd V dlh sutu fugsi erili rel dri psg u V d V yg mempuyi sift : I-. < u > = < u >. I-. < u + w > = < u > + < u w >. I-. < u > = < u > utu slr rel. I-4. < u u > d < u u > = u =.. Rug etor V yg dilegpi deg ier product < > dim rug ier product (rug hsil li dlm / rug pre-hilert). Rug ii serig ditulis deg simul : ( V < >) tu disigt deg V. c. Rug ier product yg legp yitu setip ris Cuchy rug ii oerge diseut rug Hilert. 69

28 Defiisi. (Norm) Dieri rug etor V ts ilg rel R.. Norm u dlh sutu fuggsi erili rel pd V sehigg utu setip u V mempuyi sift : N-. u. N-. u = u =. N-. N-4. u = u utu slr rel. u u +. ( etsm segitig ). Rug etor V yg dilegpi deg sutu orm dim rug erorm. Rug ii serig ditulis deg simul : ( V ) tu disigt deg V. c. Rug erorm yg legp yitu setip ris Cuchy dlm rug ii oerge diseut rug Bch. Berdsr defiisi dits Ji orm didefiisi segi sutu ier product : u = u u m : (i). Rug erorm mejdi rug ier product. (ii). Rug Bch mejdi rug Hilert. Dlm modul ii hy diperhti ier product sj sedg rug erorm rug Hilert d rug Bch tid dihs secr detil. Apil d tertri utu utu memperdlm rug-rug terseut d dpt mempeljriy dlm Alisis Fugsiol. (iii). Ji dieri etor = (... ) R m orm dri merup pjg etor yitu : = u u = u u u.... Cotoh. Misl dieri etor-etor : u = (u u ) d = ( ) R d x = (x x...x ) d y = (y y...y ) R. Didefiisi sutu fugsi < > pd R d R erturut-turut segi eriut : 7

29 < u > = u. = u + u d < x y > = x. y = x y + x y x y.. Aph < u > merup ier product dlm R.. Aph < x y > merup ier product dlm R. c. Aph rug (R < >) merup rug ier product. d. Aph rug (R < >) merup rug ier product. Jw :. Utu memperliht < u > merup ier product pd R hrus diperliht syrt (I-)-(I-4) dipeuhi. Utu setip u R diperoleh : I-. < u > = u + u = u + u ( perli ilg rel omuttif) = < u >. I-. < u + w > = < [u + ] [u + ] [w w ] > = (u + ) w + (u + ) w = (u w + w ) + (u w + w ) = (u w + u w ) + ( w + w ) = < u w > + < w >. I-. Utu sclr R : < u > = < (u u ) ( ) > = < (u u ) ( ) > = u + u = (u + u ) = < u > utu slr. I-4. < u u > = u + u d < u u > = u + u = u = d u =. u = d u =. u =. Kre (I-)-(I-4) dipeuhi m < u > merup sutu ier product pd R.. Sol ii merup geerlissi dri () sehigg peyelesiy serup. Co d selesi segi ltih. 7

30 c. Dri () diperoleh < u > merup ier product pd R. Aity rug (R < >) merup rug ier product. d. Dri () diperoleh < u > merup ier product pd R. Aity rug (R < >) merup rug ier product. Cotoh. Misl dieri etor-etor u x d y deg : x y u u = = u R d x = x y = y : : x Dieri pul mtris-mtris B d M deg : y R. B = w W = : : w : : : : w w j R +. Seljuty didefiisi sutu fugsi teroot < > pd R d R erturutturut segi eriut : < u > = u + u d < x y > = w x y + w x y w x y.. Perliht < u > merup ier product dlm R.. Perliht < x y > merup ier product dlm R. c. Perliht < u > merup ier product yg dietu oleh mtris B yitu : < u > = (B ) (Bu) = B Bu. d. Perliht < x y > merup ier product yg dietu oleh mtris W yitu : Jw : < x y > = (Wy ) (W = yw Wx.. Utu memperliht < u > merup ier product pd R hrus diperliht syrt (I-)-(I-) dipeuhi. Utu setip u R diperoleh : I-. < u > = u + u 7

31 = u + u = < u >. I-. < u + w > = < (u + u + ) (w w ) > = (u + ) w + (u + ) w = ( u w + w ) + ( u w + w ) = ( u w + u w ) + ( w + w ) = < u w > + < w >. I-. Utu slr R : < u > = < (u u ) ( ) > = < (u u ) ( ) > = u + u = ( u + u ) = < u > utu slr rel. I-4. < u u > = u + u < u u > = u + u = u = d u =. se R + d u = d u =. (Se R + ). u = (u u ) = () =. Kre (I-)-(I-4) dipeuhi m fugsi teroot < u > merup sutu ier product pd R.. Sol ii merup geerlissi dri () sehigg peyelesiy serup. Co d selesi segi ltih. u c. < u u > = < u > = u + u = ( )( u ) + ( )( u ) = [ ] u u =[ ] u u = B Bu. 7

32 d. Sol ii merup geerlissi dri (c). Co d perliht segi ltih. Cotoh. Misl dieri Rug C[] deg : C[] = { g ; g fugsi otiu pd iterl [] }. Utu setip fgh C[] didefiisi sutu fugsi < > pd C[] deg : < f g > = f ( g( dx.. Perliht hw rug (C[] < >) merup rug ier product (pre-hilert).. Ji = d = sert fugsi f( = g( = x d h( = e x Hitug ili-ili dri < f g > < f h > d < g h >. Jw :. Utu setip f g C[] diperoleh : I-. < f g > = = f ( g( dx g ( h( dx = < g f >. I-. < f + g h > = [ f ( g( ] h( dx = [ f ( h( g( h( ] dx = f ( h( dx + g ( h( dx (sift itegrl). = < f h > + < g h >. I-. Utu slr R : < f g > = f ( g( dx 74

33 = f ( g( dx = < f g >. I-4. < f f > = f ( f ( dx = < u u > = [ f ( ] dx. (se f ( x ) ) d [ f ( ] dx = f ( x ) = f( =. Kre (I-)-(I-4) dipeuhi m < > merup sutu ier product pd C[]. Aity rug ( C[] < >) merup rug ier product.. Utu = = d f( = g( = x d h( = e x diperoleh : < f g > = < g h > = x = < f h > =. dx xe x dx =.. e x dx = e d Setelh d megerti pegerti tetg ier product d rug ier product seljuty dlm modul ii disji sis husus dri rug ier product yitu sis ortogol d ortoorml. Beriut dieri defiisi tetg ortogol d sis ortoorml. Defiisi. (Ortogol d Ortoorml) Misl V merup rug ier product.. Vetor u dlm V dit ortogol ji < u > =.. Seuh himpu etor dim ortogol ji semu psg etor-etor yg ered dlm himpu terseut dlh ortogol. c. Seuh himpu etor ortogol yg setip etory mempuyi orm (pjg) stu dim ortoorml. Utu medpt sis ortogol d sis ortoorml dri sutu rug ier product d dpt meggu sutu metode ortogolissi d 75

34 ortoormlissi yg sudh sgt terel yitu proses Grm-Schmidt. Tetpi seelumy dieri eerp osep petig yg medsri proses terseut. (i). Ji V rug ier product d B = {... } sis ortoorml V m utu setip V dpt disji mejdi omisi lier : = < > + < > < >. (ii). Ji V rug ier product d {... } himpu etor ortoorml dri V sert U dlh rug yg digu oleh etor... m setip V dpt disji mejdi : = u + u deg : u = < > + < > < > d u = { < > + < > < > }. (iii). Ji V rug ier product m V mempuyi seuh sis ortoorml. Utu memperoleh sis ortoorml sutu rug ier product yg esistesiy dijmi oleh persm (iii) dpt megiuti proses ortogolissi Grm-Schmidt eriut : Misl V rug ier product deg sis B = {... }. Igi dicri sutu sis ortoorml N utu rug etor V erdsr sis B yitu : N ={... }. Lgh : Mecri yg ortoorml deg cr ompoe sis B yg pertm diut mempuyi pjg stu. =. Lgh : Mecri yg ortoorml deg. =. Lgh : Mecri yg ortoorml deg d. =. Lgh di ts diterus smpi deg lgh diperoleh : Lgh : Mecri yg ortoorml deg -. = Jdi sis ortoorml utu rug ier product V dlh : 76

35 77 N = { = =... = }. Cotoh.4 Perliht hw himpu etor-etor N = {... } dlm proses Grm- Schmidt merup etor deg orm (pjg) stu. Jw : Proses Grm-Schmidt memeri etor-etor : = = = =. = = = =. Proses diterus smpi lggh diperoleh : = = = =.

36 Cotoh.5 Dieri himpu B = { =() =() = ()} yg merup sis utu rug ier product R. Berdsr proses Grm-Schmidt dpt sis ortoorml utu R. Jw : Proses Grm-Schmidt memeri : Lgh : = =. = = ( ). Lgh : Mecri yg ortoorml deg. Dihitug =. = = ( ) ( ) ( ( ) ). = ( ) 4 ( ) = ( ) Lgh : Mecri yg ortoorml deg d. Dihitug = = d 6 =. = ( ) ( ) 6 6 ( 6 ( ) 6 ) 6 ( ( ) ) = ( ) / 4 / 4 78

37 = ( ). Jdi sis ortoorml utu rug R erdsr sis B dlh: N = { = ( ) = ( ) = ( ) } Cotoh.6 Dieri himpu B deg : B = { =() = ( ) = ( ) }. Aph B himpu ortoorml dlm R. Jw : Utu meuju B himpu ortoorml hruslh memeuhi sift : (i). < i j > = i j =. < > = (). ( ) =. < > = (). ( ) =. < > = ( ). ( ) = ½ - ½ =. (ii). j = j =. = = = / / = d = / / =. Kre (i) d (ii) dipeuhi m B himpu yg ortoorml. Cotoh.7 Dieri [ ] = {f : f poliomil derjt du pd []}. Utu setip f g [ ] didefiisi ier product pd [ ] : < f g > = f ( g( dx. 79

38 Ji B = { = = x = x } merup sis utu [ ] Dpt sis ortoorml utu [ ]. Jw : Berdsr ortogolissi Grm-Schmidt diperoleh : Lgh : = = dx = = = = = =. Lgh : Mecri yg ortoorml deg. Dihitug : = x dx = ½. x / = x / x / = ( x / ) dx = x / = = =. = (x ). = x / x / L. Lgh : Mecri yg ortoorml deg d. Dihitug : = x (x ) dx = 6 = x dx =. x x / 6 = ( x x / 6) dx = 5(6) d x x / 6 =

39 = = x x x / 6 x / 6 = x x / = 5 (6x 6x ). Jdi sis ortoorml utu rug [ ] erdsr sis B dlh: N = { = = (x ). = 5 (6x 6x ) }. Referesi Ato H.994 Elemetry Lier Ager Joh Wiley d Sos New Yor. Gryill F.A.969 Itroductio to Mtrics with Applictios i Sttistics Wdsworth Pulishig Compy Ic Cllifori. Kreyszig E. 978 Itroductory Fuctiol Alysis with Applictios Joh Wiley d Sos New Yor. Serle S.R.98 Mtrix Alger Useful for Sttistics Joh Wiley d Sos New Yor. 8