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Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 84 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 " Keywords" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 60 " Clicking on any of these words will \+ bring up its help page." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 15 " " 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "addrow" 2 "linalg[addrow]" "" } {TEXT -1 3 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "au gment" 2 "linalg[augment]" "" }{TEXT -1 1 "," }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "ditto" 2 "ditto" "" }{TEXT -1 1 "," }} {PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "dotprod" 2 "dotprod" "" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "evalm" 2 "evalm" "" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " \+ " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 2 ", " }}{PARA 15 " " 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "linsolve" 2 "linsolve" "" } {TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "mat rix" 2 "matrix" "" }{TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " \+ " }{HYPERLNK 17 "mulrow" 2 "mulrow" "" }{TEXT -1 3 ", " }}{PARA 15 " " 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "rank" 2 "linalg[rank]" "" } {TEXT -1 4 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "r estart" 2 "restart" "" }{TEXT -1 1 "," }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "rref" 2 "linalg[rref]" "" }{TEXT -1 3 ", " }} {PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "scalarmul" 2 "scalarm ul" "" }{TEXT -1 1 "," }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "swaprow" 2 "linalg[swaprow]" "" }{TEXT -1 2 ", " }}{PARA 15 "" 0 " " {TEXT -1 3 " " }{HYPERLNK 17 "vector" 2 "vector" "" }{TEXT -1 1 ", " }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "with" 2 "with" " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 " The Basic Data Types of Linear A lgebra I: Vectors" }}{PARA 0 "" 0 "" {TEXT -1 136 "Start by loading th e linear algebra package. We will use the noisy terminator to get an a ppreciation of the commands that are available:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg) ;" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,Jor danBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(a djointG%&angleG%(augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatr ixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspan G%*companionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG %(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eig envaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponenti alG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gauss elimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(h essianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)int basisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'ke rnelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(mi npolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'or thogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG %(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-sin gularvalsG%&smithG%&stackG%*submatrixG%*subvectorG%)sumbasisG%(swapcol G%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%* vecpotentG%(vectdimG%'vectorG%*wronskianG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "As you see, " }{TEXT 275 5 "MA PLE" }{TEXT -1 65 " has a rich supply of functions for dealing with l inear algebra." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "A typical vector with fou r entries is created as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v := vector( [ a, b, c, d ]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7&%\"aG%\"bG%\" cG%\"dG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Select an entry as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"cG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "You can define a general vector by specifying its dime nsion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "w := vector(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"wG-%&arrayG6$;\"\"\"\"\"&7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "You can select entries in the same way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "w[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"wG6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Various vector \+ operations are available. For example, here is how to call on the " }{HYPERLNK 17 "dotproduct" 2 "linalg[dotprod]" "" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "u := vector(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%&arrayG 6$;\"\"\"\"\"&7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotpro d(u,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%\"uG6#\"\"\"F(&%\"wGF 'F(F(*&&F&6#\"\"#F(&F*F-F(F(*&&F&6#\"\"$F(&F*F2F(F(*&&F&6#\"\"%F(&F*F7 F(F(*&&F&6#\"\"&F(&F*F " 0 "" {MPLTEXT 1 0 254 "zeroVector \+ := proc(n::posint)\n local gooseEggs, j:\n g ooseEggs := []:\n for j from 1 to n do\n goo seEggs := [op(gooseEggs), 0]:\n od:\n RETURN (vector(gooseEggs));\n end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+zeroVectorGR6#'%\"nG%'posintG6$%*gooseEggsG%\"jG6\"F-C%>8$7\" ?(8%\"\"\"F49$%%trueG>F07$-%#opG6#F0\"\"!-%'RETURNG6#-%'vectorGF;F-F-6 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "zeroVector(3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 " The Basic Data Types of Linear Algebra II: Mat rices" }}{PARA 0 "" 0 "" {TEXT -1 103 "\nRestart and reload (to clear \+ the assignments that have already been made) the linear algebra packa ge:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:\nwith(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We specify a matrix by creating a list of its r ows. Each row is a list of entries. For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "M := matrix( [ [2, 1, 3], [4, 1, 1], [3, 3, 1] ] ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7%7%\"\"#\"\"\" \"\"$7%\"\"%F+F+7%F,F,F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 28 "Notice that the syntax is " }{TEXT 259 32 "matri x( [ list1, list2, ... ] )" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "An alternative is to specify t he size of the matrix (row then column) followed by a list of entries written in the order first row, second row, ... :" }}{PARA 0 "" 0 "" {TEXT -1 34 "The syntax that is used here is " }{TEXT 277 53 "matrix ( r, c, [ entry_1, entry_2, ... , entry_r*c ] )" }{TEXT -1 9 ". Thus, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "matrix(2,3,[1,2,3,4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We can augment matrix " }{TEXT 260 1 "M" } {TEXT -1 20 " by column vector " }{TEXT 261 16 " b = [5, -1, 2] " } {TEXT -1 12 " as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "b := vector([5, -1, 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'VECTORG6#7%\"\"&!\"\"\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "M1 := augment(M,b);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'MATRIXG6#7%7&\"\"#\"\"\"\"\"$ \"\"&7&\"\"%F+F+!\"\"7&F,F,F+F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "We can perform row reduction by means of \+ the commands " }{TEXT 262 7 " addrow" }{TEXT -1 4 ", " }{TEXT 263 6 "mulrow" }{TEXT -1 9 ", and " }{TEXT 264 7 "swaprow" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "W := matrix(3,4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG-%& arrayG6%;\"\"\"\"\"$;F)\"\"%7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "swaprow(W, 2, 3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIX G6#7%7&&%\"WG6$\"\"\"F+&F)6$F+\"\"#&F)6$F+\"\"$&F)6$F+\"\"%7&&F)6$F1F+ &F)6$F1F.&F)6$F1F1&F)6$F1F47&&F)6$F.F+&F)6$F.F.&F)6$F.F1&F)6$F.F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "addrow(W, 1, 3, k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7&&%\"WG6$\"\"\"F+&F)6$F +\"\"#&F)6$F+\"\"$&F)6$F+\"\"%7&&F)6$F.F+&F)6$F.F.&F)6$F.F1&F)6$F.F47& ,&*&%\"kGF+F(F+F+&F)6$F1F+F+,&*&FAF+F,F+F+&F)6$F1F.F+,&*&FAF+F/F+F+&F) 6$F1F1F+,&*&FAF+F2F+F+&F)6$F1F4F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mulrow(W, 2, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'MATRIXG6#7%7&&%\"WG6$\"\"\"F+&F)6$F+\"\"#&F)6$F+\"\"$&F)6$F+\"\"%7&* &%\"kGF+&F)6$F.F+F+*&F7F+&F)6$F.F.F+*&F7F+&F)6$F.F1F+*&F7F+&F)6$F.F4F+ 7&&F)6$F1F+&F)6$F1F.&F)6$F1F1&F)6$F1F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Incidenta lly, " }{TEXT 274 5 "MAPLE" }{TEXT -1 50 " does not always display \+ the entries of a matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "M;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%\"MG" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "There are two ways to coax the entrie s out of it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"#\"\"\"\"\"$7%\"\"%F)F)7 %F*F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"#\"\"\"\"\"$7%\"\"%F)F)7%F*F*F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Cre ating the identity matrix can be done as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "array(identi ty,1..5,1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\" \"\"\"!F)F)F)7'F)F(F)F)F)7'F)F)F(F)F)7'F)F)F)F(F)7'F)F)F)F)F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 " Row Reduction (Reduced row echelon form) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Now l et us use the three elementary row operations to row reduce the augmen ted matrix " }{TEXT 278 2 "M1" }{TEXT -1 31 " to reduced row eche lon form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(M1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7&\"\"#\"\"\"\"\"$\"\"&7&\"\"% F)F)!\"\"7&F*F*F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "M2 := mulrow(M1, 1, 1/2); # multiplies t he first row of M1 by 1/2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%' MATRIXG6#7%7&\"\"\"#F*\"\"##\"\"$F,#\"\"&F,7&\"\"%F*F*!\"\"7&F.F.F*F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "M3 := addrow(M2, 1, 2, \+ -4); # adds -4 times row 1 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#M3G-%'MATRIXG6#7%7&\"\"\"#F*\"\"##\"\"$F,#\"\"&F,7&\"\"!!\"\"!\"&! #67&F.F.F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "M4 := addro w(M3, 1, 3, -3); # adds -3 times row 1 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M4G-%'MATRIXG6#7%7&\"\"\"#F*\"\"##\"\"$F,#\"\"&F,7& \"\"!!\"\"!\"&!#67&F2F-#!\"(F,#F5F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "M5 := mulrow(M4, 2, -1); # multiplies row 2 by -1" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M5G-%'MATRIXG6#7%7&\"\"\"#F*\"\"## \"\"$F,#\"\"&F,7&\"\"!F*F0\"#67&F2F-#!\"(F,#!#6F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "M6 := addrow(M5, 2, 1, -1/2); # adds -1/2 ti mes row 2 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M6G-%'MATRIXG 6#7%7&\"\"\"\"\"!!\"\"!\"$7&F+F*\"\"&\"#67&F+#\"\"$\"\"##!\"(F4#!#6F4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "M7 := addrow(M6, 2, 3, \+ -3/2); # adds -3/2 times row 2 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M7G-%'MATRIXG6#7%7&\"\"\"\"\"!!\"\"!\"$7&F+F*\"\"&\"#67&F+F+! #6!#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "M8 := mulrow(M7, 3 , -1/11); # multiplies row 3 by -1/11" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M8G-%'MATRIXG6#7%7&\"\"\"\"\"!!\"\"!\"$7&F+F*\"\"&\"#67&F+F+F *\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "M9 := addrow(M8, \+ 3, 1, 1); # adds 1 times row 3 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M9G-%'MATRIXG6#7%7&\"\"\"\"\"!F+!\"\"7&F+F*\"\"&\"#67&F+F+F* \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "M10 := addrow(M9, \+ 3, 2, -5); # adds -5 times row 3 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M10G-%'MATRIXG6#7%7&\"\"\"\"\"!F+!\"\"7&F+F*F+F*7&F+ F+F*\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 " Solving Systems of Linear Equati ons" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Co nsider the system of linear equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn1 := 2*x + y + 3*z = \+ 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,(%\"xG\"\"#%\"yG\"\"\" %\"zG\"\"$\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn2 := \+ 4*x + y + z = -1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,(%\"xG \"\"%%\"yG\"\"\"%\"zGF*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn3 := 3*x + 3*y + z = 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%eqn3G/,(%\"xG\"\"$%\"yGF(%\"zG\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "There are many ways to solve th is system of equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 10 "I) Using " }{TEXT 266 5 "solve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn_set := \{ eqn1, eqn2, eqn3 \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eqn_setG<%/,(%\"xG\"\"%%\"yG\"\"\"%\"zGF+!\"\"/ ,(F(\"\"$F*F0F,F+\"\"#/,(F(F1F*F+F,F0\"\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "solve(eqn_set, \{x, y, z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG!\"\"/%\"yG\"\"\"/%\"zG\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 12 "II) Using " }{TEXT 272 8 "linsolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "With the linear algebra package loaded, we do not need to type the variables. We call on " }{TEXT 268 8 "linsolve" }{TEXT -1 149 " with the coefficient matrix as the first argument and the vecto r right hand side as the second. In this case the coefficient matrix i s the matrix " }{TEXT 269 1 "M" }{TEXT -1 66 " we have been row reduc ing and the right hand side is the vector " }{TEXT 270 1 "b" }{TEXT -1 38 ". The answer is returned as a vector." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(M, b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%!\"\"\"\"\"\"\"# " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 24 "III ) Using row reduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "We have already done the row reduction. The answer we seek is the last column of the matrix that arose in the last step of \+ row reduction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "col(M10, 4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%!\"\"\"\"\"\"\"#" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Of cours e " }{TEXT 276 5 "MAPLE" }{TEXT -1 77 " provides a one-step comma nd that will do the row reduction automatically:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rref(M1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MAT RIXG6#7%7&\"\"\"\"\"!F)!\"\"7&F)F(F)F(7&F)F)F(\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "This command, " } {MPLTEXT 1 0 4 "rref" }{TEXT -1 71 ", stands for the technical termino logy for the result of the process: " }{TEXT 273 24 "row reduced eche lon form" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 " Rank" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 279 11 "Definition:" }{TEXT -1 27 " The rank of a matrix " } {XPPEDIT 19 1 "A" "6#%\"AG" }{TEXT -1 37 " is the number of leading 1 's in " }{XPPEDIT 19 1 "rref(A)" "6#-%%rrefG6#%\"AG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Exampl e: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A := matrix([ [1,1,0,1,5], [1,2,0,2,0], [3,5,0,5,5] ] );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7'\"\"\"F*\"\"!F* \"\"&7'F*\"\"#F+F.F+7'\"\"$F,F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG 6#7%7'\"\"\"\"\"!F)F)\"#57'F)F(F)F(!\"&7'F)F)F)F)F)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The rank of " } {XPPEDIT 19 1 "A" "6#%\"AG" }{TEXT -1 9 " is 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "There is also a one line command for the rank of a matrix:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rank(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Theorem:" }{TEXT -1 10 " Let " }{XPPEDIT 19 1 "A " "6#%\"AG" }{TEXT -1 13 " be an " }{XPPEDIT 19 1 "m*` x `*n" "6 #*(%\"mG\"\"\"%$~x~GF%%\"nGF%" }{TEXT -1 23 " matrix with rank " }{XPPEDIT 19 1 "r" "6#%\"rG" }{TEXT -1 8 " . Then" }}{PARA 0 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 19 1 "r <= m" "6#1%\"rG%\"mG" }{TEXT -1 10 " and " }{XPPEDIT 19 1 "r<=n" "6#1%\"rG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "b) If " }{XPPEDIT 19 1 "r=m" "6#/%\"rG %\"mG" }{TEXT -1 60 " then any system of equations with coefficient matrix " }{XPPEDIT 19 1 "A" "6#%\"AG" }{TEXT -1 60 " is consist ent. (The number of solutions is at least 1.)" }}{PARA 0 "" 0 "" {TEXT -1 8 "c) If " }{XPPEDIT 19 1 "r=n" "6#/%\"rG%\"nG" }{TEXT -1 61 " then any system of equations with coefficient matrix " } {XPPEDIT 19 1 "A" "6#%\"AG" }{TEXT -1 29 " has at most one solution ." }}{PARA 0 "" 0 "" {TEXT -1 8 "d) If " }{XPPEDIT 19 1 "r " 0 "" {MPLTEXT 1 0 31 "A1 := matrix([ [1,0], [1,0] ]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#A1G-%'MATRIXG6#7$7$\"\"\"\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "is(rank(A1) < rowdim(A1) );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "Notice that the system " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 19 1 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 19 1 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "has n o solutions and the system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 19 1 "x=1" "6#/%\"xG\"\"\"" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 19 1 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "has a unique solution. Consider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A2 := matrix([ [1,1], [1,1] \+ ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'MATRIXG6#7$7$\"\"\"F*F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "is(rank(A2) < rowdim(A 2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Notice that the system \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 19 1 "x+y=1" "6#/,&%\"xG\"\"\"%\"yGF&\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 19 1 "x+y= 1" "6#/,&%\"xG\"\"\"%\"yGF&\"\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "has infinitely many solut ions. Choose " }{XPPEDIT 19 1 "y" "6#%\"yG" }{TEXT -1 25 " arbitrari ly and take " }{XPPEDIT 19 1 "x=1-y" "6#/%\"xG,&\"\"\"\"\"\"%\"yG!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Copyright and Author \+ Information" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 54 " rowreductionR5.mws A MapleV Release 5 worksheet." } }{PARA 261 "" 0 "" {TEXT -1 26 " Author: Brian E. Blank " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 23 " Date: 26 Aug ust 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " This document may not be distributed by any medium," }}{PARA 0 " " 0 "" {TEXT -1 57 " including print, disk, and electronic transfer, \+ without" }}{PARA 0 "" 0 "" {TEXT -1 41 " prior written permission of \+ the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 51 " For more information, please contact the author:" }} {PARA 264 "" 0 "" {TEXT -1 4 " " }}{PARA 264 "" 0 "" {TEXT -1 34 " \+ Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 41 " \+ Washington University in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 28 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 35 " Telephone: (314) 935-6763" }} {PARA 265 "" 0 "" {TEXT -1 46 " e-mail: brian@math.wu stl.edu" }}{PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 " Copyright: \251 2001 Brian E. Blank, All Rights Reserved." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 1 0" 14 }{VIEWOPTS 1 1 0 3 4 1802 }