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" }}{PARA 0 "" 0 "" {TEXT 210 121 "The works heet contains several animations. They should be run by clicking the \+ plot (the initial frame), which gives the " }}{PARA 0 "" 0 "" {TEXT 210 116 "user an animation toolbar. The most useful tool is perhaps th e slider. This requires you to have Maple (or at least " }}{PARA 0 "" 0 "" {TEXT 210 19 "Maple reader (?)). " }}{PARA 0 "" 0 "" {TEXT 210 25 "There is an html-export: " }{TEXT 212 17 "ominaisarvot.html" } {TEXT 210 165 ". In this case no user interaction is possible, there \+ you can only see a continuously running animation (which after some ti me may drive you crazy, but never mind)." }}{PARA 0 "" 0 "" {TEXT 210 0 "" }}{PARA 0 "" 0 "" {TEXT 210 124 "The rest of the explanations and comments are in Finnish (for the time being). The scandinavian caract ers don't behave well " }}{PARA 0 "" 0 "" {TEXT 210 109 "in the html-e xport, that's why I use a\" for \303\244 and o\" for \303\266 (to be e asily editor-changed in the html-version)." }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT 627 10 "Alustukset" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(LinearAlg ebra):\n" }{MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "with(plots):setoptions(scaling=constrained):setoptio ns3d(axes=boxed,orientation=[-30,50]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "alias(Inv=MatrixInverse,Id=IdentityMatrix,Diag=Diagon alMatrix,Tr=Transpose):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " alias(rref=ReducedRowEchelonForm):alias(ref=GaussianElimination): \n" }{MPLTEXT 1 0 25 "alias(Id=IdentityMatrix):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 213 31 "Ma\"a\"ritella\"a\"n apufunktioita." }{TEXT 626 1 " " } }{PARA 0 "" 0 "" {TEXT 626 62 "Ovat kertaka\"ytto\"luonteisia, helpost i lennossa modifioitavia." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 22 " 1. Muotoa [xdata ]" }}{PARA 0 "" 0 "" {TEXT 626 29 " \+ [ydata ]" }}{PARA 0 "" 0 "" {TEXT 626 104 " olevan matriisin ma\"a \"ra\"a\"mien pisteiden polygonipiirto Vastaa Matlabin plot(xdata,yda ta) - komentoa." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "piirraM: =dataM->plot(convert(Transpose(dataM),listlist),axes=box,scaling=const rained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "piirraMb:=data M->plot(convert(Transpose(dataM),listlist),color=blue,axes=box): # Sam a sinisella\" (va\"rin olis voinut laittaa lisa\"argumentiksi, mutta n a\"in on va\"hemma\"n koodia.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "piirraMr:=dataM->plot(convert(Transpose(dataM),listlist),colo r=red,axes=box): # Varmuudeksi viela\" punaisella, vaikka onkin oletus ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 32 "Ylla\" olevat sopivat erila isten " }{TEXT 214 29 "alueiden kuvautumistehta\"viin" }{TEXT 626 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 106 "Seuraavaksi katsotaan yksit ta\"isten vektorien kuvautumista, erityisesti ominaisvektorihavainnoll istuksiin." }}{PARA 0 "" 0 "" {TEXT 626 8 "Funktio " }{TEXT 215 12 "pl ots[arrow]" }{TEXT 626 61 " mahdollistaa hyvin mainion helpon tavan nu olikuvioon, jossa " }{TEXT 216 10 "sinisella\"" }{TEXT 626 10 " anneta an " }{TEXT 217 13 "la\"hto\"nuoli " }{TEXT 626 3 "ja " }{TEXT 218 10 "punaisella" }{TEXT 626 5 " sen " }{TEXT 219 9 "kuvanuoli" }{TEXT 626 12 ", kas na\"in:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "kuvak ulma:=(A,u)->display(arrow(u,shape=arrow,color=blue),arrow(A.u,shape=a rrow,color=red),scaling=constrained,title=\"Lahto:sininen, kuva:punain en\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 208 "Haluamme katsoa kuvau ksen dynamiikkaa animaationa, se ka\"y taas kuin unelma. Jaetaan ta\"y skulma n:a\"a\"n osaan ja jokaista kohti piirreta\"a\"n sininen yksikk o\"vektori ja sen punainen kuva. Ajetaan animaationa (" }{TEXT 220 15 "insequence=true" }{TEXT 626 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "nuoletAni:=(A,n)->display(seq(kuvakulma(A,),Theta=[seq(kkk*2*Pi/n,kkk=0..n)]),insequence=true,scali ng=constrained):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 127 "Joskus on m ukavaa (ainakin esteettisesti) katsoa kuvaa staattisesti ja\"tta\"en k aikki nuolet kuvaan mukaan. Muuten sama, mutta \"" }{TEXT 221 16 "inse quence=false" }{TEXT 626 1 "\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "nuolikimppu:=(A,n)->display(seq(kuvakulma(A,),Theta=[seq(kkk*2*Pi/n,kkk=0..n)]),insequence=false,scaling=con strained):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 627 54 "Lineaarikuvaukse t tasossa ja niiden havannollistaminen" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 628 24 "Monikulmion kuvautuminen" }}{EXCHG {PARA 0 "" 0 "" {TEXT 626 42 "Kokeillaan ma\"a\"rittelemia\"mme funktioita:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=<<3,1>|<-2,0>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedGI(_syslibG6\"6#/I$%idGF'\")/j N<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "kolmio:=<<-1,0>|<1,0> |<0,1>|<-1,0>>; " }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protec tedGI(_syslibG6\"6#/I$%idGF'\")/pN<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "S:=A.kolmio; " }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'Mat rixG6$%*protectedGI(_syslibG6\"6#/I$%idGF'\")CqN<" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "display(array([[piirraMb(kolmio),piirraMr(S) ]]));" }}{PARA 13 "" 1 "" {TEXT 629 0 "" }{GLPLOT2D -1 -1 -1 {PLOTDATA 2 "6'-%'CURVESG6#7&7$$!#5!\"\"$\"\"!!\"\"7$$\"#5!\"\"$\"\"!! \"\"7$$\"\"!!\"\"$\"#5!\"\"7$$!#5!\"\"$\"\"!!\"\"-%+AXESLABELSG6$Q!6\" Q!6\"-%&COLORG6&%$RGBG$\"\"!!\"\"$\"\"!!\"\"$\"#5!\"\"-%*AXESSTYLEG6#% $BOXG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }}{GLPLOT2D -1 -1 -1 {PLOTDATA 2 "6'-%'CURVESG6#7&7$$!# I!\"\"$!#5!\"\"7$$\"#I!\"\"$\"#5!\"\"7$$!#?!\"\"$\"\"!!\"\"7$$!#I!\"\" $!#5!\"\"-%+AXESLABELSG6$Q!6\"Q!6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!! \"\"$\"\"!!\"\"-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 629 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "nelio:=<<0,0>|<1,0>|<1,1>|<0,1>|<0,0>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedGI(_syslibG6\"6#/I$%idGF' \");-K<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(array([[ piirraMb(nelio),piirraMr(A.nelio)]]));" }}{PARA 13 "" 1 "" {TEXT 629 0 "" }{GLPLOT2D -1 -1 -1 {PLOTDATA 2 "6'-%'CURVESG6#7'7$$\"\"!!\"\"$\" \"!!\"\"7$$\"#5!\"\"$\"\"!!\"\"7$$\"#5!\"\"$\"#5!\"\"7$$\"\"!!\"\"$\"# 5!\"\"7$$\"\"!!\"\"$\"\"!!\"\"-%+AXESLABELSG6$Q!6\"Q!6\"-%&COLORG6&%$R GBG$\"\"!!\"\"$\"\"!!\"\"$\"#5!\"\"-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#% ,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }} {GLPLOT2D -1 -1 -1 {PLOTDATA 2 "6'-%'CURVESG6#7'7$$\"\"!!\"\"$\"\"!!\" \"7$$\"#I!\"\"$\"#5!\"\"7$$\"#5!\"\"$\"#5!\"\"7$$!#?!\"\"$\"\"!!\"\"7$ $\"\"!!\"\"$\"\"!!\"\"-%+AXESLABELSG6$Q!6\"Q!6\"-%&COLORG6&%$RGBG$\"#5 !\"\"$\"\"!!\"\"$\"\"!!\"\"-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRA INEDG" 1 2 2 1 10 1 2 6 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 629 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "piirraMb(nelio);pii rraM(A.nelio); # Yksitellen piirto antaa isomman ja muokattavan kuvan \+ (tehda\"a\"n tilanteen mukaan)." }}{PARA 13 "" 1 "" {TEXT 629 0 "" } {GLPLOT2D 400 400 400 {PLOTDATA 2 "6'-%'CURVESG6#7'7$$\"\"!!\"\"$\"\"! 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#8$!0Dde98Q(=!#:7$$\"/'o#)*=o2=!#8$!0NJH&>0zi!#;7$$\"0[Wa,DW=#!#9$\"0N JH&>0zi!#;7$$\"0b9_W=n_#!#9$\"0Dde98Q(=!#:7$$\"0::d,k\"HG!#9$\"0Z\\P%* p,4$!#:7$$\"0EF'z>*p3$!#9$\"0t]c\"HzdU!#:7$$\"0-TfBOhH$!#9$\"0(**y\\zE e`!#:7$$\"0l(za%)H`M!#9$\"/p[(*)RUP'!#97$$\"0h@T4+gb$!#9$\"079UF'o*G(! #:7$$\"0z4x[@Eg$!#9$\"0Z\\P%*p,4)!#:7$$\"0-N$)QFCf$!#9$\"0kQ/!o1j()!#: 7$$\"06MIcyb_$!#9$\"0^#))e[w(H*!#:7$$\"/crd#HJS$!#8$\"0J'G6;$eo*!#:7$$ \"0/s5d5qA$!#9$\"0yWJ,Z6#**!#:7$$\"#I!\"\"$\"#5!\"\"7$$\"0#[\"ojxcs#!# 9$\"0yWJ,Z6#**!#:-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%+A XESLABELSG6$Q!6\"Q!6\"-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG " 1 2 2 1 10 1 2 0 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 629 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 57 "Ympyra\"n kuva na\"ytta\"a\" e llipsilta\", onkohan na\"in aina?" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 628 13 "Strangin talo" }}{PARA 0 "" 0 "" {TEXT 210 75 "Gilbert Strang: An Introduction to Linear Algebra, kts, myo\"s MIT-luentoja)" }{TEXT 246 0 "" }}{PARA 0 "" 0 "" {TEXT 626 129 "Ympyro\"iden, kolmioiden, ne lio\"iden ym. lisa\"ksi \"Strangin talo\" on hyva\" esimerkkikuvio lin aarikuvausten havainnollistamisessa. " }}{PARA 0 "" 0 "" {TEXT 626 42 "(Muita kuuluisia on mm. \"Arnoldin kissa\".)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "talo:=Tr(<<0,0,-1,6,13,12,12,3,3,6,6,0>|<0,9,8,15 ,8,9,0,0,5,5,0,0>>):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "#pi irraMb(talo);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(ar ray([[piirraMb(talo),piirraMr(A.talo)]]));" }}{PARA 13 "" 1 "" {TEXT 629 0 "" }{GLPLOT2D -1 -1 -1 {PLOTDATA 2 "6'-%'CURVESG6#7.7$$\"\"!!\" \"$\"\"!!\"\"7$$\"\"!!\"\"$\"#!*!\"\"7$$!#5!\"\"$\"#!)!\"\"7$$\"#g!\" \"$\"$]\"!\"\"7$$\"$I\"!\"\"$\"#!)!\"\"7$$\"$?\"!\"\"$\"#!*!\"\"7$$\"$ ?\"!\"\"$\"\"!!\"\"7$$\"#I!\"\"$\"\"!!\"\"7$$\"#I!\"\"$\"#]!\"\"7$$\"# g!\"\"$\"#]!\"\"7$$\"#g!\"\"$\"\"!!\"\"7$$\"\"!!\"\"$\"\"!!\"\"-%+AXES LABELSG6$Q!6\"Q!6\"-%&COLORG6&%$RGBG$\"\"!!\"\"$\"\"!!\"\"$\"#5!\"\"-% *AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }}{GLPLOT2D -1 -1 -1 {PLOTDATA 2 "6'-%'CU RVESG6#7.7$$\"\"!!\"\"$\"\"!!\"\"7$$!$!=!\"\"$\"\"!!\"\"7$$!$!>!\"\"$! #5!\"\"7$$!$?\"!\"\"$\"#g!\"\"7$$\"$I#!\"\"$\"$I\"!\"\"7$$\"$!=!\"\"$ \"$?\"!\"\"7$$\"$g$!\"\"$\"$?\"!\"\"7$$\"#!*!\"\"$\"#I!\"\"7$$!#5!\"\" $\"#I!\"\"7$$\"#!)!\"\"$\"#g!\"\"7$$\"$!=!\"\"$\"#g!\"\"7$$\"\"!!\"\"$ \"\"!!\"\"-%+AXESLABELSG6$Q!6\"Q!6\"-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!! \"\"$\"\"!!\"\"-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 2 1 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 629 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT 628 49 "Geometrisin kuvauksin annettuja \+ lineaarikuvauksia" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 31 "1. 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!!\"\"-%&STYLEG6#%,PATCHNOGRIDG-%&TITLEG6$-%)_TYPESETG6#Q=Lahto:sinine n,~kuva:punainen6\"-%-TRANSPARENCYG6#$\"\"!!\"\"-%%VIEWG6$;$!#5!\"\"$ \"#5!\"\";$!#5!\"\"$\"#5!\"\"-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 0 0 4 1 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" "Curve 1" "Curve 2" " Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" " Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curv e 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22 " "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "C urve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Cu rve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve \+ 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Cur ve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 7 3" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79" " Curve 80" }}{TEXT 629 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 121 "No ei sitten milla\"a\"n voi olla ominaisarvoja, nuolet pyo\"riva\"t sam aa vauhtia, eiva\"tka\" missa\"a\"n mene pa\"a\"llekka\"in. " }}{PARA 0 "" 0 "" {TEXT 626 101 "Mutta mutta ... meha\"n sallimme myo\"s kompl eksiset skalaarit. Kompleksikertlaskuhan on tason kiertoa!" }}{PARA 0 "" 0 "" {TEXT 626 12 "Lasketaanpa!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "karpoly:=det(A-lambda*Id(2));\n" }{MPLTEXT 1 0 3 " " }}{PARA 11 "" 1 "" {XPPMATH 20 ",(\"\"\"F#*&\"\"##F#F%I'lambdaG6\"F# !\"\"*$F'F%F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lam:=solve (karpoly=0,lambda);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*$\"\"##\" \"\"F%F&*&^##!\"\"F%F'F%F&F',&F$F&*&^#F&F'F%F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "#Eigenvectors(A);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "M1:=A-lam[1]*Id(2);rM1:=ref(M1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedGI(_syslibG6\"6#/I$%idGF'\")_^ 1@" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedGI(_syslibG 6\"6#/I$%idGF'\")#*f1@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x :='x';t:='t';\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"tG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x2:=t: x1:=solve(rM1[1,1]*x+rM1[1,2]*x2=0,x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&^#!\"\"\"\"\"I\"tG6\"F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "v:=subs(t=1,);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$% idGF(\")o\"GJ(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "lam[1];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$\"\"##\"\"\"F$F%*&^##!\"\"F$F&F$F %F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v;A;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF( 6#/I$%idGF(\")o\"GJ(" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*pr otectedGI(_syslibG6\"6#/I$%idGF'\")_zkK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A.v,map(evalc,lam[1].v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF)6#/I $%idGF)\")sq1@-F$6#/F.\")7t1@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 626 96 "Samoja ovat, joten oikein laskettiin. Toinen ominaisarvo/vektori saad aan kompleksikonjugoimalla." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 530 7 "Ky symys" }{TEXT 626 117 ": Onko nyt niin, etta\" joko nuolet meneva\"t j ossain pa\"a\"llekka\"in (reaaliset ominaisarvot) tai kyseessa\" on ki erto. " }}{PARA 0 "" 0 "" {TEXT 531 10 "Vastaus: " }{TEXT 626 92 "Kyl la\", paitsi kiertoon voi yhdistya\" venytys/kutistus (kertominen reaa lisella skalaarilla)." }}{PARA 0 "" 0 "" {TEXT 532 24 "Onko ta\"ssa\" \+ siis kaikki" }{TEXT 626 52 ", mita\" 2 x 2-matriisesita voidaan yleise sti sanoa? " }{TEXT 533 15 "Eipa\" olekaan! " }}{PARA 0 "" 0 "" {TEXT 626 90 "Kukaan ei ole sanonut, etta\" ominaisvektoreita on aina kaksi \+ lineaarisesti riippumatonta. 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