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}} {SECT 0 {PARA 18 "" 0 "" {TEXT 256 36 " LINEAR TRANSFORMATIONS OF THE \+ PLANE" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 22 "Linear Algebra Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 168 "To understand the concept of linear transformation; to become familiar with various types of transformations, such as dil ations, rotations, reflections and projections." }}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 0 "" }{TEXT 276 23 "Background Information:" }}{PARA 0 "" 0 "" {TEXT -1 62 "When a 2 x 2 matrix A multiplies a 2 x 1 \+ column vector " }{TEXT 277 1 "v" }{TEXT -1 45 ", the result is anothe r 2 x 1 column vector " }{TEXT 278 1 "w" }{TEXT -1 27 ". We say that \+ the vector " }{TEXT 279 1 "v" }{TEXT -1 10 " has been " }{TEXT 280 11 "transformed" }{TEXT -1 17 " into the vector " }{TEXT 282 1 "w" } {TEXT -1 12 ", that is, A" }{TEXT 283 1 "v" }{TEXT -1 5 " = " } {TEXT 284 1 "w" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 243 "If we imagine that A multiplies all ve ctors in the plane, we get a transformation of the whole plane. Certa in matrices will do predictable things to every vector. For example, i f A is a diagonal matrix with both diagonal entries equal to " } {TEXT 328 1 "r" }{TEXT -1 56 ", it will dilate (stretch) every vector \+ by a factor of " }{TEXT 329 1 "r" }{TEXT -1 5 ", if " }{TEXT 330 2 " \+ r" }{TEXT -1 14 " > 1. If 0 < " }{TEXT 331 2 "r " }{TEXT -1 81 "< 1, \+ every vector will \"shrink.\" In both cases, the direction of the vect or will " }{TEXT 285 3 "not" }{TEXT -1 26 " change. (What happens if \+ " }{TEXT 332 3 " r " }{TEXT -1 10 "= 1? If " }{TEXT 333 2 "r " } {TEXT -1 10 "= 0? If " }{TEXT 334 1 "r" }{TEXT -1 146 " < 0?) [Note: a diagonal matrix is one in which the diagonal entries from upper lef t to lower right are non-zero, and all other entries are zero.]" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "A very im portant theorem you will learn states the following:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 10 "The first " }{TEXT 304 6 "column" }{TEXT 305 105 " of a transformation matrix will be wha tever that transformation does to the vector <1, 0>. The second " } {TEXT 306 6 "column" }{TEXT 307 72 " of a transformation matrix will b e whatever that transformation does to" }{TEXT -1 1 " " }{TEXT 287 19 "the vector <0, 1>." }{TEXT -1 171 " (Does this seem plausible for th e dilation matrix discussed above?) This theorem will often make it v ery easy to determine the matrix of a transformation when we have a " }{TEXT 312 21 "geometric description" }{TEXT -1 33 " of what the trans formation does." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 281 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 " " }{TEXT 259 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 138 "a) U se the theorem above to find the matrix of the transformation that rot ates every vector counterclockwise 30 degrees around the origin." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "b) Show a plot of the \"unit square\" before and after the above rotation is ap plied" }{MPLTEXT 0 21 0 "" }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 9 "Solution:" }{TEXT 262 0 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 288 0 "" }{TEXT 289 48 "a) Recall that the components of a vector are <" }{TEXT 337 1 "r " }{TEXT 338 5 " cos " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 0 "" }{TEXT 290 2 ", " }{TEXT 291 1 "r" }{TEXT 339 5 " sin " } {XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 ">" }{TEXT 292 0 "" } {TEXT 293 8 ", where " }{TEXT 335 2 " r" }{TEXT 336 38 " is the magni tude of the vector and " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 " " }{TEXT 294 0 "" }{TEXT 295 45 " is the angle that the vector \+ makes with the " }{TEXT 313 8 "positive" }{TEXT 314 62 " x-axis. Using the theorem, <1, 0> will be transformed into " }{TEXT 297 1 "<" } {XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 2 "/ " }{TEXT 309 1 "2" }{TEXT -1 2 ", " }{TEXT 308 3 "1/2" }{TEXT -1 2 "> " }{TEXT 298 0 "" }{TEXT 299 38 " and <0, 1> will be transformed into" }} {PARA 4 "" 0 "" {TEXT 340 3 " <" }{TEXT 301 0 "" }{TEXT 300 6 "-1/2, \+ " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 2 "/ " } {TEXT 310 0 "" }{TEXT 311 1 "2" }{TEXT -1 1 ">" }{TEXT 302 0 "" } {TEXT 303 51 ". So, the matrix of this transformation will be " } {XPPEDIT 18 0 "MATRIX([[sqrt(3)/2, -1/2], [1/2, sqrt(3)/2]]);" "6#-%'M ATRIXG6#7$7$*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\",$*&F-F-F.F/F/7$*&F-F-F.F /*&-F*6#F,F-F.F/" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 275 1 " " }{TEXT 296 68 "b) We activate t he Maple linear algebra, plots and lamp* packages: " }{TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "with (linalg); with (plots) : with (Lamp);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonal G%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addco lG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG%&basisG %'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG %)colspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG%*crosspro dG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotpr odG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermatrix G%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+forwardsub G%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)ha damardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*indexfu ncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)ja cobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG% 'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normal izeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatrixG %+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rr efG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*submatrixG%*subv ectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG%* transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#7_p%'Axis3dG%*BackidmatG%*BacksolveG% (BandmatG%*BasisgridG%$BugG%&Bug3dG%'Bug3dhG%)CharpolyG%+Chevron3dhG%& ClockG%*ColorlistG%.ComponentplotG%)CopyintoG%*CrossprodG%'Cube3dG%(Cu be3dhG%$DetG%(DiagmatG%(DotprodG%*DrawlinesG%+DrawmatrixG%+DrawplanesG %(DrawvecG%*Drawvec3dG%&EvalmG%(EvaluesG%)EvectorsG%'ExpandG%*Genmatri xG%)GetcolorG%%GridG%)GridgameG%*GridvectsG%)HeadtailG%(Hotel3dG%)Hote l3dhG%&HouseG%)House3dhG%&IdmatG%(InverseG%&Jet3dG%'Jet3dhG%*Jordanmat G%(Lamp3dhG%&LamphG%(LetterLG%(LetterNG%(LetterSG%(LetterXG%$MagG%)Map colorG%,MappicturesG%/MatrixtovectorG%)MatsolveG%&MovieG%*NullbasisG%% PathG%(ProjectG%+ProjectmatG%-Projectmat3dG%)QuestionG%(RandmatG%'Redu ceG%+ReflectmatG%-Reflectmat3dG%*ResidualsG%*RotatematG%,Rotatemat3dG% &RowopG%+SymbollistG%+TrajectoryG%-Trajectory3dG%*TransformG%-Translat ematG%/Translatemat3dG%)UnitspanG%*VandermatG%+VectorgridG%+Vectorline G%-VectranslateG%*XshearmatG%*YshearmatG" }}}{PARA 0 "" 0 "" {TEXT -1 220 "[Note: if you highlight any of the \"Lamp\" commands above and go to the help screen, you will find an explanation of what the command \+ does, the proper format for entering it, and executable examples of it s use. Try this!]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 253 "We will use the \"Unitspan\" command to show a ni ce picture of the unit square. Then we will use \"matrix\" ( a regular Maple command) and \"Drawmatrix\" and \"Transform\" (\"Lamp\" command s) to show how we can rotate the unit square 30 degrees counterclockwi se. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v := matrix(2,1,[1, 0]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'matrixG6#7$7#\"\"\" 7#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "w := matrix(2,1,[0,1]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'matrixG6#7$7#\"\"!7#\"\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Unitspan(v ,w);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 243 187 187 {PLOTDATA 2 "6+-%)POL YGONSG6%7%7$$\"\"\"\"\"!$F*F*7$$\"+++++()!#5$\"++++gG!#67$F-$!++++gGF2 -%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%&STYLEG6#%&PATCHG-F$6&7$7$F-F+7$ F+F+-F76&F9F*F*F*-%*THICKNESSG6#\"\"#-F>6#%%LINEG-%%TEXTG6%7$$\"++++I6 !\"*F+Q#~u6\"-%&COLORG6&F9F*F*$\"\"&!\"\"-F$6%7%7$F+F(7$F4F-7$F0F--F76 &F9F:F+F+F=-F$6&7$7$F+F-FEFFFHFL-FP6%7$F+FSQ#~vFWFX-F$6$7&FEF'7$F(F(F[ o-F76&F9F:F:F+-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%'NORMALG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 44.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "M := ma trix(2,5,[0,0,1,1,0,0,1,1,0,0]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"MG-%'matrixG6#7$7'\"\"!F*\"\"\"F+F*7'F*F+F+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Drawmatrix(M);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 263 181 181 {PLOTDATA 2 "6*-%)POLYGONSG6&7'7$$\"\"!F)F(7$F($ \"\"\"F)7$F+F+7$F+F(F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%'SYMBOLG6# %'CIRCLEG-%&STYLEG6#%&POINTG-%'CURVESG6$7$F'F*F/-F?6$7$F*F-F/-F?6$7$F- F.F/-F?6$7$F.F'F/-F$6%7#F'-F06&F2F,F,F,F:-%(SCALINGG6#%,CONSTRAINEDG-% *AXESSTYLEG6#%'NORMALG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "N := matrix([[sqrt(3 )/2, -1/2], [1/2, sqrt(3)/2]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"NG-%'matrixG6#7$7$,$*&\"\"#!\"\"\"\"$#\"\"\"F,F0#F-F,7$F/F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Transform(N,M);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 237 215 215 {PLOTDATA 2 "60-%)POLYGONSG6'7'7$$\" \"!F)F(7$F($\"\"\"F)7$F+F+7$F+F(F'-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F( -%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-%*THICKNESSG6#F,-%'CURVESG6%7 $F'F*F/F>-FB6%7$F*F-F/F>-FB6%7$F-F.F/F>-FB6%7$F.F'F/F>-F$6&7#F'-F06&F2 F,F,F,F:F>-F$6'7'F'7$$!+++++]!#5$\"+SSDg')FY7$$\"+SSDgOFY$\"+/a-m8!\"* 7$FZ$\"+++++]FYF'-F06&F2F(F(F3F6F:-F?6#\"\"$-FB6%7$F'FVF_oFao-FB6%7$FV FfnF_oFao-FB6%7$FfnF\\oF_oFao-FB6%7$F\\oF'F_oFao-F$6&FPFQF:Fao-%(SCALI NGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%'NORMALG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 43.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "* Written permission for the use \+ of Linear Algebra Modules Project (LAMP) package from the authors, Her man, et. al. and publisher Addison-Wesley on file." }}}}{PARA 0 "" 0 " " {TEXT -1 77 "_______________________________________________________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 0 "" }{TEXT 263 0 "" }{TEXT 264 10 "ASSIGNMENT" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 10 "Problem 1:" } {TEXT 266 0 "" }{TEXT 267 0 "" }{TEXT 268 0 "" }{TEXT 269 0 "" }{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "a) Use the same technique as in the \+ solved example to find the matrix of the transformation that " }{TEXT 315 8 "reflects" }{TEXT -1 522 " every vector across the line through \+ the origin that makes a 30-degree angle with the positive x-axis. (Thi s line is of infinite length and lies in quadrants I and III.) To refl ect a vector across a line means the following: From the head of the v ector draw a perpendicular to the line, creating a line segment. Conti nue this segment a distance equal to its own length. The point where y ou are now is the head of the reflected vector. Remember, find what wo uld happen to <1 ,0> and <0, 1> under this transformation. " }} {PARA 0 "" 0 "" {TEXT -1 48 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 67 "b) Show a plot of the unit squ are before and after the reflection. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 273 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 97 "a) Use the same technique as in the solved example to find the mat rix of the transformation that " }{TEXT 316 8 "projects" }{TEXT -1 14 " every vector " }{TEXT 317 4 "onto" }{TEXT -1 277 " the line through \+ the origin that makes a 30-degree angle with the positive x-axis. (Thi s line is of infinite length and lies in quadrants I and III.) To proj ect a vector onto a line means the following: From the head of the vec tor draw a perpendicular to the line. Stop there." }}{PARA 0 "" 0 "" {TEXT -1 155 " The point where you are now is the head of the proje cted vector. Remember, find what would happen to <1, 0> and <0, 1> \+ under this transformation. " }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 200 "b) Show a plot of the unit square before and after the projection . \+ " }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 274 20 "Problem 3 (Summar y):" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "(Somewhat of a cha llenge.) Generalize what you did in part a) of each problem above, inc luding the solved example. That is, find " }{TEXT 318 5 "three" } {TEXT -1 66 " GENERAL matrices: one that rotates every vector counterc lockwise " }{TEXT 319 0 "" }{TEXT 320 5 "theta" }{TEXT 321 0 "" } {TEXT -1 76 " radians; one that reflects every vector across the line \+ making an angle of " }{TEXT 322 0 "" }{TEXT 323 5 "theta" }{TEXT 324 0 "" }{TEXT -1 103 " radians with the positive x-axis; and one that pr ojects every vector onto the line making an angle of " }{TEXT 325 0 " " }{TEXT 326 5 "theta" }{TEXT 327 0 "" }{TEXT -1 35 " radians with the positive x-axis. " }}}{PARA 0 "" 0 "" {TEXT -1 77 "__________________ ___________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 96 "MSIP Grant #P120A80089-98: \"Three Urban Calcul us Reform Programs: Adopting the Best\" 1998-2001" }}}{MARK "7 2 14 \+ 1 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }