{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(plots);with(Lin earAlgebra);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7Z%(animateG%*animate3 dG%-animatecurveG%&arrowG%-changecoordsG%,complexplotG%.complexplot3dG %*conformalG%,conformal3dG%,contourplotG%.contourplot3dG%*coordplotG%, coordplot3dG%-cylinderplotG%,densityplotG%(displayG%*display3dG%*field plotG%,fieldplot3dG%)gradplotG%+gradplot3dG%,graphplot3dG%-implicitplo tG%/implicitplot3dG%(inequalG%,interactiveG%-listcontplotG%/listcontpl ot3dG%0listdensityplotG%)listplotG%+listplot3dG%+loglogplotG%(logplotG %+matrixplotG%(odeplotG%'paretoG%,plotcompareG%*pointplotG%,pointplot3 dG%*polarplotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.po lyhedraplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoptio ns3dG%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplo tG%+textplot3dG%)tubeplotG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7br%#&xG %$AddG%(AdjointG%3BackwardSubstituteG%+BandMatrixG%&BasisG%-BezoutMatr ixG%/BidiagonalFormG%-BilinearFormG%5CharacteristicMatrixG%9Characteri sticPolynomialG%'ColumnG%0ColumnDimensionG%0ColumnOperationG%,ColumnSp aceG%0CompanionMatrixG%0ConditionNumberG%/ConstantMatrixG%/ConstantVec torG%%CopyG%2CreatePermutationG%-CrossProductG%-DeleteColumnG%*DeleteR owG%,DeterminantG%)DiagonalG%/DiagonalMatrixG%*DimensionG%+DimensionsG %+DotProductG%6EigenConditionNumbersG%,EigenvaluesG%-EigenvectorsG%&Eq ualG%2ForwardSubstituteG%.FrobeniusFormG%4GaussianEliminationG%2Genera teEquationsG%/GenerateMatrixG%2GetResultDataTypeG%/GetResultShapeG%5Gi vensRotationMatrixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3Hermit ianTransposeG%/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/Id entityMatrixG%2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimil arG%*IsUnitaryG%2JordanBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompos itionG%-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%2MatrixEx ponentialG%/MatrixFunctionG%.MatrixInverseG%5MatrixMatrixMultiplyG%+Ma trixNormG%,MatrixPowerG%5MatrixScalarMultiplyG%5MatrixVectorMultiplyG% 2MinimalPolynomialG%&MinorG%(ModularG%)MultiplyG%,NoUserValueG%%NormG% *NormalizeG%*NullSpaceG%3OuterProductMatrixG%*PermanentG%&PivotG%*Popo vFormG%0QRDecompositionG%-RandomMatrixG%-RandomVectorG%%RankG%6Rationa lCanonicalFormG%6ReducedRowEchelonFormG%$RowG%-RowDimensionG%-RowOpera tionG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-ScalarVectorG%*Schur FormG%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVectorG%)SumBasisG% 0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG%0TridiagonalFor mG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,VectorAngleG%5VectorM atrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+ZeroMatrixG%+ZeroV ectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "#section 1 .9: commands are provided which picture the effects of transformation s on the unit square." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "in equal(x+y<1,x=-5..5,y=-5..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6&-%)POLYGONSG6$7%7$$!\"%\"\"!$\"\"&F*7$F+F+7$F+F(-%& COLORG6&%$RGBG$\"#?!\"#F3F3-F$6$7'7$$!\"&F*F:7$F:F+F-7$F+F:F9-F06&F2$ \"#qF5F@$\"#5!\"\"-%'CURVESG6$7$F.F'-%*LINESTYLEG6#\"\"#-%&STYLEG6#%,P ATCHNOGRIDG" 1 3 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "testplot:=A->inequal([(A^(-1).Vector([x, y]))[1]<1,(A^(-1).Vector([x,y]))[2]<1,(A^(-1).Vector([x,y]))[1]>0,(A^( -1).Vector([x,y]))[2]>0],x=-5..5,y=-5..5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)testplotGf*6#%\"AG6\"6$%)operatorG%&arrowGF(-%(inequ alG6%7&2&-%\".G6$*&\"\"\"F69$!\"\"-%'VectorG6#7$%\"xG%\"yG6#F6F62&F26# \"\"#F62\"\"!F12FEFA/F=;!\"&\"\"&/F>FHF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "anglematrix:=theta->Matrix([[cos(theta),-sin(the ta)],[sin(theta),cos(theta)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%, anglematrixGf*6#%&thetaG6\"6$%)operatorG%&arrowGF(-%'MatrixG6#7$7$-%$c osG6#9$,$-%$sinGF3!\"\"7$F6F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "anglematrix(Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'RTABLEG6%\"(/5(e-%'MATRIXG6#7$7$,$*&\"\"#!\"\"F.#\"\"\"F.F1,$*&F.F /F.F0F/7$F,F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "t estplot(%);" }}{PARA 13 "" 1 "" {GLPLOT2D 240 240 240 {PLOTDATA 2 "6&- %)POLYGONSG6'7%7$$!0\"piPky&e$!#9$\"\"&\"\"!7$F+F+7$F+F(7%7$$!\"&F-F(7 $F2F+7$$\"0\"piPky&e$F*F+7%7$F2F2F47$F+F27%F9F.F:-%&COLORG6&%$RGBG$\"# ?!\"#F@F@-F$6$7'F9F4F.F:F9-F=6&F?$\"#qFBFH$\"#5!\"\"-%'CURVESG6'7$F'F/ 7$F5F17$F4F:7$F9F.-%*LINESTYLEG6#\"\"#-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "testplot(ang lematrix(Pi/6));" }}{PARA 13 "" 1 "" {GLPLOT2D 240 240 240 {PLOTDATA 2 "6&-%)POLYGONSG6'7&7$$!0()ov!30KF;7$FBFC- %*LINESTYLEG6#\"\"#-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "testplot(anglematrix(Pi/6).M atrix([[2,0],[0,1]]));" }}{PARA 13 "" 1 "" {GLPLOT2D 240 240 240 {PLOTDATA 2 "6&-%)POLYGONSG6'7%7$$!0D'*=p-Nx&!#:$\"\"&\"\"!7$F+F+7$F+$ !0SWy.a-m%!#97&7$$!\"&F-$!0()ov!30K7$F5F@7$F+FC7$F+F5-%&COLORG 6&%$RGBG$\"#?!\"#FMFM-F$6$7'F>F9F.FHF>-FJ6&FL$\"#qFOFU$\"#5!\"\"-%'CUR VESG6'7$F'F/7$F4F:7$FBF?7$FFFG-%*LINESTYLEG6#\"\"#-%&STYLEG6#%,PATCHNO GRIDG" 1 3 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "t estplot(Matrix([[2,0],[0,1]]).anglematrix(Pi/6));" }}{PARA 13 "" 1 "" {GLPLOT2D 496 496 496 {PLOTDATA 2 "6&-%)POLYGONSG6'7%7$$!0vP^h,TY$!#9$ \"\"&\"\"!7$F+F+7$F+$!0?A*=q7IBF*7&7$$!\"&F-$!07[fM^n)G!#:7$F4F+F.7$F+ $\"0J`8@w!)f#F*7&7$F4F47$F4$\"0?A*=q7IVF*7$F+$!0?A*=q7IVF*7$F+F47&F>7$ F4$!01uHnvLW\"F*7$F+$\"01uHnvLW\"F*FE-%&COLORG6&%$RGBG$\"#?!\"#FQFQ-F$ 6$7'F>F9F.FEF>-FN6&FP$\"#qFSFY$\"#5!\"\"-%'CURVESG6'7$F'F/7$F3F:7$F?FB 7$FGFJ-%*LINESTYLEG6#\"\"#-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "# Exercise 1: do prob lems 4-10 even (2 doesn't work), section 1.9, p. 90, using the tools h ere. Use matrices that represent the rotations and reflections mentio ned in the problems in your calculations." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "?LUDecomposition" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "(X,Y,Z):=LUDecomposition(Matrix([[2,1,3],[-1,3,4],[3, 3,5]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"XG%\"YG%\"ZG6%-%'RTA BLEG6%\")o+gK-%'MATRIXG6#7%7%\"\"\"\"\"!F37%F3F2F37%F3F3F2%'MatrixG-F* 6%\")3]$\\$-F.6#7%F17%#!\"\"\"\"#F2F37%#\"\"$F@#FC\"\"(F2F6-F*6%\")wK \\K-F.6#7%7%F@F2FC7%F3#FEF@#\"#6F@7%F3F3#!#8FEF6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABL EG6%\")o+gK-%'MATRIXG6#7%7%\"\"\"\"\"!F-7%F-F,F-7%F-F-F,%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "X.Y.Z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")Gm,N-%'MATRIXG6#7%7%\"\"#\"\"\"\"\"$7%! \"\"F.\"\"%7%F.F.\"\"&%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rref:=ReducedRow EchelonForm;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rrefG%6ReducedRowEc helonFormG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "# Exercise 2 (for LU factorizations): do problem 31, section 2.5, p. 150-1. In p articular, do parts a and c (b is optional). Notice that the" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "#matrices in the LU decompo sition are band matrices and the inverse of the matrix A is not a band matrix. Try changing entries in the matrix and observe that this con tinues to work. Notice that I have typed in the matrix from the probl em." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "A:=Matrix([[4,-1,-1 ,0,0,0,0,0],[-1,4,0,-1,0,0,0,0],[-1,0,4,-1,-1,0,0,0],[0,-1,-1,4,0,-1,0 ,0],[0,0,-1,0,4,-1,-1,0],[0,0,0,-1,-1,4,0,-1],[0,0,0,0,-1,0,4,-1],[0,0 ,0,0,0,-1,-1,4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG 6%\"(c^R&-%'MATRIXG6#7*7*\"\"%!\"\"F/\"\"!F0F0F0F07*F/F.F0F/F0F0F0F07* F/F0F.F/F/F0F0F07*F0F/F/F.F0F/F0F07*F0F0F/F0F.F/F/F07*F0F0F0F/F/F.F0F/ 7*F0F0F0F0F/F0F.F/7*F0F0F0F0F0F/F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "# \+ Exercise 3 (determinants): Try problems 18, 19, chapter 3 supplemental , p. 213." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Determinant(Ma trix([[1,2,3],[4,6,6],[7,8,9]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#! #7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# column space and null space bases" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "# Exercise 4: problem 16, page 243: use our stand ard technique using rref" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# to find the basis. Then try using the built-in ColumnSpace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# function to find a basis a s well. Is the relation between" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# the two bases obvious? If it isn't, show that all \+ elements" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "of one of the \+ bases can be expressed in terms of the other basis (you can choose whi ch one to work from). i.e., show that each element of one of the base s is in the span of the other basis. I provide a convenient matrix!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "A:=Transpose(<<1,-2,6,5, 0>|<0,1,-1,-3,3>|<0,-1,2,3,-1>|<1,1,-1,-4,1>>); #notice that this not ation for matrices allows us to write them column by column -- so I ha d to transpose, since I wrote it row by row!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"(O2Y&-%'MATRIXG6#7&7'\"\"\"!\"#\"\" '\"\"&\"\"!7'F2F.!\"\"!\"$\"\"$7'F2F4\"\"#F6F47'F.F.F4!\"%F.%'MatrixG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(Oqh&-%'MATRIXG6#7&7'\"\"\"\"\"!F-! \"\"!\"#7'F-F,F-!\"$\"\"&7'F-F-F,F-\"\"#7'F-F-F-F-F-%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ColumnSpace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%-%'RTABLEG6%\"(KKh&-%'MATRIXG6#7&7#\"\"\"7#\" \"!F.F,&%'VectorG6#%'columnG-F%6%\"(#zfa-F)6#7&F.F,F.7#!\"\"F0-F%6%\"( wKR&-F)6#7&F.F.F,7#!\"%F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "#Exercise 5: Do the same for null spaces with problem 10, sectio n 4.3, p. 243. First use our usual method using rref, then use the bu ilt-in NullSpace command, then verify that all elements of one" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "#basis can be expressed in t erms of the other basis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A:=<<1,-2,0>|<0,1,2>|<-5,6,-8>|<1,-2,1>|<4,-2,9>>;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")'z'=K-%'MATRIXG6#7%7'\"\"\"\"\"! !\"&F.\"\"%7'!\"#F.\"\"'F3F37'F/\"\"#!\")F.\"\"*%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")W0PK-%'MATRIXG6#7%7'\"\"\"\"\"!!\"&F-\" \"(7'F-F,!\"%F-\"\"'7'F-F-F-F,!\"$%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "NullSpace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#< $-%'RTABLEG6%\")?sOK-%'MATRIXG6#7'7#\"\"&7#\"\"%7#\"\"\"7#\"\"!F2&%'Ve ctorG6#%'columnG-F%6%\")W7UK-F)6#7'7#!\"(7#!\"'F27#\"\"$F0F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "#this is from the first demo -- this time you get an assignment..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "C:=Matrix([[1,2,3],[-1,0,2],[3,3,1]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6%\")Cs`K-%'MATRIXG6#7%7%\"\"\" \"\"#\"\"$7%!\"\"\"\"!F/7%F0F0F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "MatrixInverse(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6%\");:eK-%'MATRIXG6#7%7%\"\"'!\"(!\"%7%F-\"\")\"\"&7%\"\" $!\"$!\"#%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "?RowO peration;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "E:=Matrix([C, IdentityMatrix(3)]); #notice that a list of matrices will be put toge ther in a sensible way by the Matrix command." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6%\"(/!=_-%'MATRIXG6#7%7(\"\"\"\"\"#\" \"$F.\"\"!F17(!\"\"F1F/F1F.F17(F0F0F.F1F1F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "rop:=RowOperation; #abbreviate the RowOp eration command." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ropG%-RowOperat ionG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "?RowOperation" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "#the following function cons tructs elementary matrices." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "em:=(n,x,y)->if y=0 then rop(IdentityMatrix(n),x) else rop(Identit yMatrix(n),x,y) end if;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#emGf*6%% \"nG%\"xG%\"yG6\"6$%)operatorG%&arrowGF*@%/9&\"\"!-%$ropG6$-_%.LinearA lgebraG%/IdentityMatrixG6#9$9%-F36%F5F;F0F*F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 96 "#the following development shows how to use Ma ple to find the inverse of E using row operations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "# I put in extra steps to find elementary m atrices for each row operation; this is why I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "# use %% instead of % -- this looks for the nex t-to-last result of a calculation instead of the last result (if I use d % I would apply the row operation to the elementary matrix from the \+ previous step, which is not what I want)." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 2 "E;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\" (/!=_-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F,\"\"!F/7(!\"\"F/F-F/F,F/7(F.F.F ,F/F/F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rop(%,[ 2,1],1); E1:=em(3,[2,1],1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA BLEG6%\"(!Q)G&-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F,\"\"!F/7(F/F-\"\"&F,F, F/7(F.F.F,F/F/F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G-%' RTABLEG6%\"(7UP&-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F.F.F/7%F/F/F.%'MatrixG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rop(%%,[3,1],-3); E2:=e m(3,[3,1],-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(/OU&- %'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F,\"\"!F/7(F/F-\"\"&F,F,F/7(F/!\"$!\")F 3F/F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G-%'RTABLEG6%\" (O%=b-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%!\"$F/F.%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rop(%%,2,1/2); E3:=em(3,2,1 /2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"([Ce&-%'MATRIXG6 #7%7(\"\"\"\"\"#\"\"$F,\"\"!F/7(F/F,#\"\"&F-#F,F-F3F/7(F/!\"$!\")F5F/F ,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G-%'RTABLEG6%\"(3%R b-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/#F.\"\"#F/7%F/F/F.%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rop(%%,[3,2],3); E4:=em(3,[3 ,2],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"('4Cc-%'MATRI XG6#7%7(\"\"\"\"\"#\"\"$F,\"\"!F/7(F/F,#\"\"&F-#F,F-F3F/7(F/F/#!\"\"F- #!\"$F-#F.F-F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E4G-%'RT ABLEG6%\"(S3p&-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/\"\"$F.%'Matrix G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rop(%%,3,-2); E5:=em(3 ,3,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(wOx&-%'MATRI XG6#7%7(\"\"\"\"\"#\"\"$F,\"\"!F/7(F/F,#\"\"&F-#F,F-F3F/7(F/F/F,F.!\"$ !\"#%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E5G-%'RTABLEG6%\"( w![e-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/!\"#%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "rop(%%,[2,3],-5/2); 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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")OgMK-%'MATRIXG6#7%7 %\"\"'!\"(!\"%7%F-\"\")\"\"&7%\"\"$!\"$!\"#%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "E8.E7.E6.E5.E4.E3.E2.E1; #notice that t he inverse is the product of the elementary matrices I constructed. H ow could you express C itself as a product of elementary matrices?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(+wC&-%'MATRIXG6#7%7%\" \"'!\"(!\"%7%F-\"\")\"\"&7%\"\"$!\"$!\"#%'MatrixG" }}}{PARA 11 "" 1 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "C.%; #the product of C with this matrix is the identity as expected." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(w-F&-%'MATRIXG6#7%7%\"\"\"\" \"!F-7%F-F,F-7%F-F-F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "# Exercise 6: follow the model work above to express Matrix( [[1,2,3],[4,6,6],[7,8,9]]) as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "a product of elementary matrices (hint: you can cut-and-paste o n the work above)." }}}}{MARK "32 0 0" 192 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 5871004 32600068 34935008 32493276 35016628 5395156 5460736 5617036 5613232 5459792 5393276 32186796 32370544 32367220 32421244 32537224 32581516 5218004 5288380 5374212 5423604 5518436 5582448 5539408 5624096 5690840 5773676 5848076 5801408 5853572 32288424 32340788 32572328 5414332 32346036 5247600 5270276 }{RTABLE M7R0 I4RTABLE_SAVE/5871004X,%)anythingG6"6"[gl!"%!!!#%"#"#,$*$""##"""F)F*F',$F(#!""F )F'F& } {RTABLE M7R0 I5RTABLE_SAVE/32600068X/%)anythingG6"6"[gl#"!!!!#*"$"$$$!"#$!!!!!!"#$!!!!!!"""! !!!!!F& } {RTABLE M7R0 I5RTABLE_SAVE/34935008X,%)anythingG6#&%+triangularG6#%&lowerG6"[gl!")!!!#'"$"$" ""#!""""##""$F.F+#F0""(F+F* } {RTABLE M7R0 I5RTABLE_SAVE/32493276X,%)anythingG6"6"[gl!"%!!!#*"$"$""#""!F("""#""(F'F(""$#"# 6F'#!#8F+F& } {RTABLE M7R0 I5RTABLE_SAVE/35016628X,%)anythingG6"6"[gl!"%!!!#*"$"$""#!""""$"""F)F)F)""%""&F & } {RTABLE M7R0 I4RTABLE_SAVE/5395156X,%)anythingG6"6"[gl!"%!!!#[o")")""%!""F(""!F)F)F)F)F(F'F) F(F)F)F)F)F(F)F'F(F(F)F)F)F)F(F(F'F)F(F)F)F)F)F(F)F'F(F(F)F)F)F)F(F(F'F)F(F)F)F )F)F(F)F'F(F)F)F)F)F)F(F(F'6" } {RTABLE M7R0 I4RTABLE_SAVE/5460736X,%)anythingG6"6"[gl!"%!!!#5"%"&"""""!F(F'!"#F'!""F'""'F*" "#F*""&!"$""$!"%F(F/F*F'6" } {RTABLE M7R0 I4RTABLE_SAVE/5617036X,%)anythingG6"6"[gl!"%!!!#5"%"&"""""!F(F(F(F'F(F(F(F(F'F( !""!"$F(F(!"#""&""#F(6" } {RTABLE M7R0 I4RTABLE_SAVE/5613232X*%)anythingG6"6"[gl!#%!!!"%"%"""""!F(F'6" } {RTABLE M7R0 I4RTABLE_SAVE/5459792X*%)anythingG6"6"[gl!#%!!!"%"%""!"""F'!""6" } {RTABLE M7R0 I4RTABLE_SAVE/5393276X*%)anythingG6"6"[gl!#%!!!"%"%""!F'"""!"%6" } {RTABLE M7R0 I5RTABLE_SAVE/32186796X,%)anythingG6"6"[gl!"%!!!#0"$"&"""!"#""!F)F'""#!"&""'!") 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