{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 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 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 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 {PARA 18 "" 0 "" {TEXT 256 36 "BASIS AND DIMENSION OF VECTOR S PACES" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 28 "Linear Algebra Maple Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Objective:" }} {PARA 0 "" 0 "" {TEXT -1 135 "To understand the concepts of basis and \+ dimension of a vector space; to find a basis for and determine a basis of a given vector space." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 259 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 36 "a) Find \+ a basis for the subspace of " }{XPPEDIT 18 0 "R^5;" "6#*$%\"RG\"\"&" } {TEXT -1 67 " spanned by (2,0,3,0,1), (0,-1,0,0,0), (0,1,0,3,0) and (0 ,0,0,2,0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "b) What is the dimension of this subspace?" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 9 "Solution:" }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with (linalg); with (plots): with (Lamp);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlo ckG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjoint G%&angleG%(augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(c harmatG%)charpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*com panionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delc olsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvalu esG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'e xtendG%,ffgausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG% *gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessian G%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG %(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG% *laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG %'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG% *permanentG%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratf ormG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularv alsG%&smithG%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcol G%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%* vecpotentG%(vectdimG%'vectorG%*wronskianG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7_p%'Axis3dG%*BackidmatG%*BacksolveG%(BandmatG%*Basisgr idG%$BugG%&Bug3dG%'Bug3dhG%)CharpolyG%+Chevron3dhG%&ClockG%*ColorlistG %.ComponentplotG%)CopyintoG%*CrossprodG%'Cube3dG%(Cube3dhG%$DetG%(Diag matG%(DotprodG%*DrawlinesG%+DrawmatrixG%+DrawplanesG%(DrawvecG%*Drawve c3dG%&EvalmG%(EvaluesG%)EvectorsG%'ExpandG%*GenmatrixG%)GetcolorG%%Gri dG%)GridgameG%*GridvectsG%)HeadtailG%(Hotel3dG%)Hotel3dhG%&HouseG%)Hou se3dhG%&IdmatG%(InverseG%&Jet3dG%'Jet3dhG%*JordanmatG%(Lamp3dhG%&Lamph G%(LetterLG%(LetterNG%(LetterSG%(LetterXG%$MagG%)MapcolorG%,Mappicture sG%/MatrixtovectorG%)MatsolveG%&MovieG%*NullbasisG%%PathG%(ProjectG%+P rojectmatG%-Projectmat3dG%)QuestionG%(RandmatG%'ReduceG%+ReflectmatG%- Reflectmat3dG%*ResidualsG%*RotatematG%,Rotatemat3dG%&RowopG%+Symbollis tG%+TrajectoryG%-Trajectory3dG%*TransformG%-TranslatematG%/Translatema t3dG%)UnitspanG%*VandermatG%+VectorgridG%+VectorlineG%-VectranslateG%* XshearmatG%*YshearmatG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v 1:= matrix(5,1,[2,0,3,0,1]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v 1G-%'matrixG6#7'7#\"\"#7#\"\"!7#\"\"$F+7#\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "v2:= matrix(5,1,[0,-1,0,0,0]);\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#v2G-%'matrixG6#7'7#\"\"!7#!\"\"F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v3:= matrix(5,1,[0,1,0,3,0]) ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v3G-%'matrixG6#7'7#\"\"!7#\" \"\"F)7#\"\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v4:= mat rix(5,1,[0,0,0,2,0]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v4G-%'ma trixG6#7'7#\"\"!F)F)7#\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "M:=augment(v1,v2,v3,v4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"MG-%'matrixG6#7'7&\"\"#\"\"!F+F+7&F+!\"\"\"\"\"F+7&\"\"$F+F+F+7&F+F+ F0F*7&F.F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M1:=gauss elim(M);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7'7&\" \"#\"\"!F+F+7&F+!\"\"\"\"\"F+7&F+F+\"\"$F*7&F+F+F+F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Backsolve(M1);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6%\")Oe#*>-%'MATRIXG6#7%7#\"\"!7##\"\"#\"\" $F-&%'VectorG6#%'columnG" }}}{PARA 0 "" 0 "" {TEXT -1 56 "This means t hat the four vectors are linearly dependent." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "N:= augment(v1,v2,v3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7'7%\"\"#\"\"!F+7%F+!\"\"\"\"\"7%\"\" $F+F+7%F+F+F07%F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "N1 := gausselim(N);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N1G-%'matrixG 6#7'7%\"\"#\"\"!F+7%F+!\"\"\"\"\"7%F+F+\"\"$7%F+F+F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Backsolve(N1);\n" }}{PARA 8 "" 1 " " {TEXT -1 35 "Error, (in Backsolve) no solutions\n" }}}{PARA 0 "" 0 " " {TEXT -1 26 "This means that the three " }{XPPEDIT 19 1 "v[1];" "6#& %\"vG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[2];" "6#&%\"vG6#\"\" #" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[3];" "6#&%\"vG6#\"\"$" }{TEXT -1 35 " vectors are linearly independent." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "P:= augment(v1,v2,v4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7'7%\"\"#\"\"!F+7%F+!\"\"F+7%\"\"$F+F +7%F+F+F*7%\"\"\"F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "P1 := gausselim(P);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G-%'matrixG 6#7'7%\"\"#\"\"!F+7%F+!\"\"F+7%F+F+F*7%F+F+F+F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "Backsolve(P1);\n" }}{PARA 8 "" 1 "" {TEXT -1 35 "Error, (in Backsolve) no solutions\n" }}}{PARA 0 "" 0 "" {TEXT -1 35 "This means that the three vectors " }{XPPEDIT 19 1 "v[1];" "6#&% \"vG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[2];" "6#&%\"vG6#\"\"# " }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[4];" "6#&%\"vG6#\"\"%" }{TEXT -1 40 " are linearly independent. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R:= augment(v2,v3,v4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG-%'matrixG6#7'7%\"\"!F*F*7%!\"\"\"\"\"F*F)7%F*\" \"$\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "R1:= gausseli m(R);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R1G-%'matrixG6#7'7%!\"\" \"\"\"\"\"!7%F,\"\"$\"\"#7%F,F,F,F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Backsolve(R1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'RTABLEG6%\");f#*>-%'MATRIXG6#7$7##\"\"#\"\"$F+&%'VectorG6#%'columnG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "This means that the vectors \+ " }{XPPEDIT 19 1 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[3];" "6#&%\"vG6#\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[4]; " "6#&%\"vG6#\"\"%" }{TEXT -1 27 " are linearly dependent. " }} {PARA 0 "" 0 "" {TEXT -1 27 "Hence we can select either " }{XPPEDIT 19 1 "v[1];" "6#&%\"vG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[2]; " "6#&%\"vG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 19 1 "v[3];" "6#&%\"vG6 #\"\"$" }{TEXT -1 4 " or " }{XPPEDIT 19 1 "v[1];" "6#&%\"vG6#\"\"\"" } {TEXT -1 2 ", " }{XPPEDIT 19 1 "v[2];" "6#&%\"vG6#\"\"#" }{TEXT -1 2 " , " }{XPPEDIT 19 1 "v[4];" "6#&%\"vG6#\"\"%" }{TEXT -1 70 " as the bas is of the subspace, and the dimension of the subspace is 3." }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "_________ ________________________________________________________________" }} {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 269 0 "" }{TEXT 270 0 "" }{TEXT 271 0 "" }{TEXT 272 0 "" }{TEXT 273 0 "" }{TEXT 274 0 "" } {TEXT 275 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "a) Find a basis for the \+ subspace of " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 37 " c ontained in the plane 2x-3y+4z = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 42 "b) What is the dimension of this subspac e?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }{TEXT 267 0 "" } {TEXT 268 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 276 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 158 "a) Given the set \{(2,0,2 ), (0,4,0)\}. What condition(s) would a third vector (to be added to t he set) have to satisfy in order that the new set be a basis of " } {XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "b) Give an example of \+ a vector satisfying the condition(s) you found in part a)." }}{PARA 0 "" 0 "" {TEXT -1 89 " \+ " }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 0 "" }{TEXT 277 20 "Problem 3 (Summary):" }}{PARA 0 "" 0 " " {TEXT -1 37 "Give the simplest possible basis of " }{XPPEDIT 18 0 " R^n;" "6#)%\"RG%\"nG" }{TEXT -1 29 ". What is the dimension of " } {XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 2 " ?" }}}{PARA 0 "" 0 "" {TEXT -1 77 "______________________________________________________ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 94 "MSIP Grant #P1 20A80089-98: \"Three Urban Calculus Reform Programs: Adopting the Best \" 1998-2001" }}}{MARK "11 4 3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 19925836 19925916 }{RTABLE M7R0 I5RTABLE_SAVE/19925836X*%)anythingG6"6"[gl!#%!!!"$"$""!#""#""$F(6" } {RTABLE M7R0 I5RTABLE_SAVE/19925916X*%)anythingG6"6"[gl!#%!!!"#"##""#""$F'6" }