{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 24 "Simple Vector Operations " }}{PARA 19 "" 0 "" {TEXT -1 17 "February 10, 1998" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "This covers the materia l from page 138 to page 146 of the text," }}{PARA 0 "" 0 "" {TEXT 256 26 "Elementary Linear Algerbra" }{TEXT -1 9 " by Hill." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Load the linear Algebra Package. It cont ains many of the vector" }}{PARA 0 "" 0 "" {TEXT -1 11 "operations." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%( augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG %(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG %+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG% 'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG %*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%' mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG %&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG% 'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith G%&stackG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylv esterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%(vectd imG%'vectorG%*wronskianG" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Elem entary Vector Operations." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Enter a few vectors for later use." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a \+ := vector([1,1,1]);\nb := vector([-2,5,-3]);\nc := vector([1,-4,-4]); " }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"aG-%'VECTORG6#7%\"\"\"F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"b G-%'VECTORG6#7%!\"#\"\"&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG -%'VECTORG6#7%\"\"\"!\"%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The distance between points is calculated using the" }{TEXT 257 1 " " } {HYPERLNK 17 "norm" 2 "linalg,norm" "" }{TEXT -1 10 " function." }} {PARA 0 "" 0 "" {TEXT -1 65 "Notice the 2 in the command. Can you tel l what the norm command " }}{PARA 0 "" 0 "" {TEXT -1 23 "does if you o mit the 2?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "pt1 := [1,0];\npt2 := [2, -2];\nv := pt1-pt2;\nnorm(v,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ pt1G7$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pt2G7$\"\"#!\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7$!\"\"\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*$\"\"&#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The dotproduct is also easy to calculate with the " } {HYPERLNK 17 "dotprod" 2 "dotprod" "" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(a ,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Using the dotproduct it is easy to calculate a project ion. Here is the projection of" }}{PARA 0 "" 0 "" {TEXT 258 1 "c" } {TEXT -1 6 " onto " }{TEXT 259 1 "a" }{TEXT -1 32 ". What happens if \+ you omit the " }{TEXT 260 5 "evalm" }{TEXT -1 1 "?" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "evalm((dotprod(c,a)/dotprod(a,a))* a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%#!\"(\"\"$F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 " Exercises" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Are any of the vector s " }{TEXT 265 1 "a" }{TEXT -1 2 ", " }{TEXT 266 1 "b" }{TEXT -1 5 " a nd " }{TEXT 267 1 "c" }{TEXT -1 33 " orthogonal to any of the others? " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Find the projection of " } {TEXT 261 1 "b" }{TEXT -1 6 " onto " }{TEXT 262 1 "c" }{TEXT 268 1 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Find two orthogonal vectors t hat have the same span as " }{TEXT 263 1 "a" }{TEXT -1 5 " and " } {TEXT 264 1 "c" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Are there constants r, s, and t such that r" }{TEXT 269 1 "a" }{TEXT -1 4 " + s" }{TEXT 270 1 "b" }{TEXT -1 4 " + t" }{TEXT 271 1 "c" } {TEXT -1 12 " = [1,2,-1]?" }}}}}{MARK "3 2 0 0" 19 }{VIEWOPTS 1 1 0 1 1 1803 }