{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 35 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "" 0 0 102 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 0 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 263 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" 263 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 0 1 0 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 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 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 0 } {CSTYLE "" 263 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 270 "Times" 0 0 102 0 230 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 20 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 275 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 "Hea ding 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Time s" 1 12 128 0 0 1 1 1 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 "List Item" -1 14 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 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 3 "" 0 "" {TEXT -1 43 "Introduction to matrices and line ar systems" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B. C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearA lgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "Maple data structures for vectors and mat rices - new and old" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 42 "Versions of Maple from 6 onwards have a \+ \" " }{TEXT 259 3 "new" }{TEXT -1 2 "\" " }{HYPERLNK 17 "rtable" 2 "rt able" "" }{TEXT -1 22 "-based data structured" }{TEXT 259 1 " " } {TEXT -1 69 "to handle Matrices and Vectors, which was not available w ith Maple 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 15 "older structure" }{TEXT -1 29 " is still av ailable, however." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "The 2 " }{TEXT 274 1 "x" }{TEXT -1 10 " 2 matrix " } {XPPEDIT 18 0 "A=matrix([[2,-1],[3,7]])" "6#/%\"AG-%'matrixG6#7$7$\"\" #,$\"\"\"!\"\"7$\"\"$\"\"(" }{TEXT -1 48 " can be entered as follows. Note the use of an " }{TEXT 259 10 "upper case" }{TEXT -1 1 " " } {TEXT 0 1 "M" }{TEXT -1 13 " in the word " }{TEXT 0 6 "Matrix" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Matrix([[2,-1],[3,7]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*Cq]]\"-%'MATRIXG6#7$7$\"\"#!\"\"7$\"\"$\"\"(%'Mat rixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " If you select the last output, and " }{TEXT 270 38 "view the text fiel d in the context bar" }{TEXT -1 31 ", you will see something like: " } {TEXT 275 22 "Matrix(%id = 35873600)" }{TEXT -1 151 ", although the nu mber 35873600 will be different. The number (probably) gives the addre ss in memory where the data structure for the matrix is located." }} {PARA 0 "" 0 "" {TEXT -1 217 "Each time this worksheet is opened a dif ferent number appears within the square brackets on the next command l ine and this number automatically coincides with the value which appea rs when the last output is selected. " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Matrix(%id = 150507024); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")))G(H\"-%'MATRIXG6# 7$7$\"\"#!\"\"7$\"\"$\"\"(%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "older type" }{TEXT -1 51 " matrix data structure is constructed if you use a " }{TEXT 259 10 "lower case" }{TEXT -1 1 " " }{TEXT 0 1 "m" }{TEXT -1 13 " in t he word " }{TEXT 0 6 "matrix" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "matrix([[2,- 1],[3,7]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"#! \"\"7$\"\"$\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "If you select this output, and " }{TEXT 270 38 "view the \+ text field in the context bar" }{TEXT -1 60 ", you will see exactly wh at is entered on the command line. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 90 "With both data structures, you can acces s a particular matrix entry by way of two indices." }}{PARA 0 "" 0 "" {TEXT -1 15 "Given a matrix " }{TEXT 288 1 "A" }{TEXT -1 36 ", in math ematical notation we write " }{XPPEDIT 18 0 "A[i,j]" "6#&%\"AG6$%\"iG% \"jG" }{TEXT -1 31 " for the entry which is in the " }{TEXT 267 1 "i" }{TEXT -1 12 " th row and " }{TEXT 268 1 "j" }{TEXT -1 14 " th column \+ of " }{TEXT 289 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "A = matrix([[7, -2, 3], [-1, 8,4], [6, -5, 9]])" "6#/%\"AG-%'matrixG6#7%7% \"\"(,$\"\"#!\"\"\"\"$7%,$\"\"\"F-\"\")\"\"%7%\"\"',$\"\"&F-\"\"*" } {TEXT -1 8 ", then " }{XPPEDIT 18 0 "A[3,2] = -5;" "6#/&%\"AG6$\"\"$ \"\"#,$\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A[ 3,2]" "6#&%\"AG6$\"\"$\"\"#" }{TEXT -1 32 " is entered on a command li e as " }{TEXT 260 6 "A[3,2]" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 57 ": Matrices \+ can be entered on a command line by using the " }{TEXT 259 14 "matrix \+ palette" }{TEXT -1 30 " available from the view menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A := Matr ix([[7,-2,3],[-1,8,4],[6,-5,9]]);\nA[3,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*#fq2:-%'MATRIXG6#7%7%\"\"(!\"#\"\" $7%!\"\"\"\")\"\"%7%\"\"'!\"&\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# !\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 " Individual entries of a matrix can be assigned values by assignment st atements of the form " }{TEXT 0 15 "A[i,j] := ?????" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A[3,2] := `Hello`;\nA;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"A G6$\"\"$\"\"#%&HelloG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$ \"*#fq2:-%'MATRIXG6#7%7%\"\"(!\"#\"\"$7%!\"\"\"\")\"\"%7%\"\"'%&HelloG \"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "It is possible to convert between the old and new matrix data structu res using " }{TEXT 0 18 "convert(..,matrix)" }{TEXT -1 5 " and " } {TEXT 0 18 "convert(..,Matrix)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Try selecting the two oup uts from . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(A,matrix);\nconvert(%,Matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"(!\"#\"\"$7%!\"\"\"\")\" \"%7%\"\"'%&HelloG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6 $\"*GA>\\\"-%'MATRIXG6#7%7%\"\"(!\"#\"\"$7%!\"\"\"\")\"\"%7%\"\"'%&Hel loG\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Vectors can be entered using the command " }{TEXT 0 6 "Vector" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Both row and column vect ors are handled. A " }{TEXT 259 13 "column vector" }{TEXT -1 23 " is j ust the same as a " }{TEXT 259 22 "matrix with one column" }{TEXT -1 8 ", and a " }{TEXT 259 10 "row vector" }{TEXT -1 43 " is just the sam e as a matrix with one row." }}{PARA 0 "" 0 "" {TEXT -1 49 "The defaul t is to construct a column vector . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Vector([1,2,3,4]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*)[gM:-%'MATRIXG6#7&7#\" \"\"7#\"\"#7#\"\"$7#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " . . . but you can construct a row vector by givi ng " }{TEXT 0 6 "Vector" }{TEXT -1 10 " the word " }{TEXT 0 3 "row" } {TEXT -1 6 " as a " }{TEXT 259 9 "subscript" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Ve ctor[row]([1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$ \"*'pQY:-%'VECTORG6#7&\"\"\"\"\"#\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "A single entry or " }{TEXT 259 9 "component" }{TEXT -1 47 " of a vector can accessed using a single i ndex." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "V := Vector[row]([x,x^2,x^3,x^4,x^5]);\nV[4];\nV[4] : = V[4] -1;\nV;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'RTABLEG6$\" *!)>1^\"-%'VECTORG6#7'%\"xG*$)F-\"\"#\"\"\"*$)F-\"\"$F1*$)F-\"\"%F1*$) F-\"\"&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG\"\"%\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"VG6#\"\"%,&*$)%\"xGF'\"\"\"F,F,! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*!)>1^\"-%'VECTO RG6#7'%\"xG*$)F+\"\"#\"\"\"*$)F+\"\"$F/,&*$)F+\"\"%F/F/F/!\"\"*$)F+\" \"&F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Shortcuts \+ for entering vectors and matrices " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 72 "Versions of Maple f rom 6 onwards have notational shortcuts for creating " }{HYPERLNK 17 " rtable" 2 "rtable" "" }{TEXT -1 46 "-based Matrices and Vectors in the help page: " }{HYPERLNK 17 "LinearAlgebra,General,MVshortcut" 2 "Line arAlgebra,General,MVshortcut" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "This help page is reproduced below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT 275 11 "< a, b, c >" }{TEXT -1 25 " constructs an object by " }{TEXT 259 4 "rows" }{TEXT -1 2 ". " }} {PARA 15 "" 0 "" {TEXT 275 13 "< a | b | c >" }{TEXT -1 25 " construct s an object by " }{TEXT 259 7 "columns" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "If any of the values given between the matching pair of angled brackets " }{TEXT 35 3 "< > " }{TEXT -1 10 " is not a " }{HYPERLNK 17 "scalar" 2 "type,scalar" "" }{TEXT -1 124 ", (A scalar is anything which is not a vector or matri x), then a Matrix is constructed. Otherwise, a Vector is constructed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 9 "Examp les:" }{TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 25 "Construct a column Vector" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "< 1, 2, 3 >;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(3#*3%-%'MATRIXG6#7%7#\" \"\"7#\"\"#7#\"\"$" }}}{PARA 5 "" 0 "" {TEXT -1 22 "Construct a row Ve ctor" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "< 1 | 2 | 3 >;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(G&*3%-%'VECTORG6#7%\"\" \"\"\"#\"\"$" }}}{PARA 5 "" 0 "" {TEXT -1 29 "Construct a Matrix by co lumns" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "< < 1, 2, 3 > | < 4 , 5, 6 > >;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(Wa@%-%'M ATRIXG6#7%7$\"\"\"\"\"%7$\"\"#\"\"&7$\"\"$\"\"'" }}}{PARA 5 "" 0 "" {TEXT -1 40 "Augment this Matrix with a column Vector" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "< % | < x, y, z > >;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(k+E%-%'MATRIXG6#7%7%\"\"\"\"\"%%\"xG7%\" \"#\"\"&%\"yG7%\"\"$\"\"'%\"zG" }}}{PARA 5 "" 0 "" {TEXT -1 26 "Constr uct a Matrix by rows" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "< < \+ 1 | 2 | 3 >, < 4 | 5 | 6 > >;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RT ABLEG6$\"(O,J%-%'MATRIXG6#7$7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'" }}} {PARA 5 "" 0 "" {TEXT -1 12 "Construct a " }{TEXT 35 5 "1 x n" }{TEXT -1 26 " Matrix from 2 row Vectors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "< < 1 | 2 | 3 > | < 4 | 5 | 6 > >;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6$\"([tN%-%'MATRIXG6#7#7(\"\"\"\"\"#\"\"$\" \"%\"\"&\"\"'" }}}{PARA 14 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 229 "In all cases, the object is constructed by using default values for all construction parameters other than the entries (and or ientation, in the case of Vectors). In particular, the dimension infor mation is deduced from the data. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 34 "Operations on matrices using Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A := Matrix( [[2,-1],[3,7]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$ \"(;OL(-%'MATRIXG6#7$7$\"\"#!\"\"7$\"\"$\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Warning" }{TEXT -1 32 ": \+ If you replace the upper case " }{TEXT 0 1 "M" }{TEXT -1 13 " in the w ord " }{TEXT 0 6 "Matrix" }{TEXT -1 19 " with a lower case " }{TEXT 0 1 "m" }{TEXT -1 12 ", and write " }{TEXT 0 6 "matrix" }{TEXT -1 38 ", \+ this gives the older data structure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "B := Matrix([[-4, 5],[1, 3 ]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\"(#Hcz-%'MA TRIXG6#7$7$!\"%\"\"&7$\"\"\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 83 "Matrix addition, subtraction and multipli cation by a scalar can be performed using " }{TEXT 0 3 "+,-" }{TEXT -1 5 " and " }{TEXT 0 1 "*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "C := 3*A-B;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6$\"(Kam(-%'MATRIXG6#7 $7$\"#5!\")7$\"\")\"#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Matrix multiplication is denoted by the symbol \"" } {TEXT 262 1 "." }{TEXT -1 4 "\" . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E := A . B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6$\")G[$p#-%'MATRIXG6#7$7$!\"*\"\" (7$!\"&\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 7 "inverse" }{TEXT -1 73 " of a non-singular sq uare matrix can be calculated using the exponent -1." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A := Matri x([[2, -1], [3, 7]]);\nAinv := A^(-1);\nAinv . A;\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"(O-Z(-%'MATRIXG6#7$7$\"\"#!\"\"7$ \"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%AinvG-%'RTABLEG6$\"(o ,W(-%'MATRIXG6#7$7$#\"\"(\"#<#\"\"\"F07$#!\"$F0#\"\"#F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(ss^(-%'MATRIXG6#7$7$\"\"\"\"\"!7$ F-F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " Both negative and positive exponents are allowed, since for example, \+ " }{TEXT 0 3 "A^3" }{TEXT -1 23 " is the matrix product " }{TEXT 0 5 " A.A.A" }{TEXT -1 6 ", and " }}{PARA 0 "" 0 "" {TEXT 0 6 "A^(-3)" } {TEXT -1 26 " is the inverse matrix of " }{TEXT 0 3 "A^3" }{TEXT -1 31 " or, equivalently, the product " }{TEXT 0 20 "A^(-1).A^(-1).A^(-1) " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "A^(-3);\nAinv^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(Kde(-%'MATRIXG6#7$7$#\"$&H\"%8\\#\"#kF.7$#!$#>F.# !#DF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(Wxc(-%'MATRIXG 6#7$7$#\"$&H\"%8\\#\"#kF.7$#!$#>F.#!#DF." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "Suggestion" }{TEXT -1 75 ": Try q uestion 1 in the tasks section before proceding to the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Solving linear system s of equations with Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 29 "A system of linear equations:" }} {PARA 258 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "2*x[1]-4*x[2]-x[3] = -4;" "6#/,(*&\"\"#\"\"\"&%\"xG6#F'F'F'*&\"\"%F'&F)6#F&F'!\"\"&F)6#\" \"$F/,$F,F/" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x[1] +x[2]-2*x[3] = 8;" "6#/,(*&\"\"$\"\"\"&%\"xG6#F'F'F'&F)6#\"\"#F'*&F-F' &F)6#F&F'!\"\"\"\")" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x[1]-x[2]+x[3] = 1;" "6#/,(*&\"\"$\"\"\"&%\"xG6#F'F'F'&F)6#\"\"#! \"\"&F)6#F&F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "can be solved by means of the " }{TEXT 0 5 "solve" }{TEXT -1 9 " command." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "x := 'x':\nsys := \{2*x[1]-4*x[2]-x[3]=-4,3*x[1]+x[2]-2*x[3]=8, 3*x[1]-x[2]+x[3]=1\};\nsoln := solve(sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG<%/,(&%\"xG6#\"\"\"\"\"$&F)6#\"\"#!\"\"&F)6#F,F+ F+/,(F(F/*&\"\"%F+F-F+F0F1F0!\"%/,(F(F,F-F+*&F/F+F1F+F0\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG<%/&%\"xG6#\"\"$#!#@\"#?/&F(6#\"\"\" #\"#`\"#S/&F(6#\"\"##\"#xF4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 47 ": The equations must be giv en to the procedure " }{TEXT 0 5 "solve" }{TEXT -1 40 " as a set, desi gnated by curly brackets " }{TEXT 0 2 "\{\}" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "We can check th e solutions by substituting back into the original equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(soln,sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/!\"%F%/\"\") F'/\"\"\"F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "This system of equations can be converted to a matrix equation \+ of the form: " }{XPPEDIT 18 0 "A*`.`*x = b" "6#/*(%\"AG\"\"\"%\".GF&% \"xGF&%\"bG" }{TEXT -1 9 ", namely," }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "matrix([[2, -4, -1], [3, 1, -2], [3, -1, 1]])*matrix ([[x[1]], [x[2]], [x[3]]]) = matrix([[-4], [8], [1]]);" "6#/*&-%'matri xG6#7%7%\"\"#,$\"\"%!\"\",$\"\"\"F-7%\"\"$F/,$F*F-7%F1,$F/F-F/F/-F&6#7 %7#&%\"xG6#F/7#&F:6#F*7#&F:6#F1F/-F&6#7%7#,$F,F-7#\"\")7#F/" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "A = matrix([[2, -4, -1], [3, 1, -2], [ 3, -1, 1]]);" "6#/%\"AG-%'matrixG6#7%7%\"\"#,$\"\"%!\"\",$\"\"\"F-7%\" \"$F/,$F*F-7%F1,$F/F-F/" }{TEXT -1 8 " is the " }{TEXT 259 18 "coeffic ient matrix" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple package " }{TEXT 0 13 "LinearAlgebra " }{TEXT -1 22 " contains a procedure " }{TEXT 0 14 "GenerateMatrix" } {TEXT -1 84 " which will perform this task automatically, except that \+ the rows may be rearranged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "x := 'x':\nwith(LinearAlgebra):\ns ys := [2*x[1]-4*x[2]-x[3] = -4,\n 3*x[1]+x[2]-2*x[3] = 8,\n \+ 3*x[1]-x[2]+x[3] = 1]:\nvar := [x[1],x[2],x[3]]:\n(A,b) := Gener ateMatrix(sys,var);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"AG%\"bG6$ -%'RTABLEG6$\"*#fC]:-%'MATRIXG6#7%7%\"\"#!\"%!\"\"7%\"\"$\"\"\"!\"#7%F 5F3F6-F)6$\"*ol2\\\"-F-6#7%7#F27#\"\")7#F6" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 45 "The equations must be given to the procedure " }{TEXT 0 14 "GenerateM atrix" }{TEXT -1 41 " as a list designated by square brackets " } {TEXT 0 2 "[]" }{TEXT -1 42 ", rather than a set, as was the case with " }{TEXT 0 5 "solve" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 145 "The last statement makes use of a feature in Maple of making a \"doub le assignment\", for both the coeffiicient matrix and the vector of co nstants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "LinearSolve" }{TEXT -1 14 ", also in \+ the " }{TEXT 0 13 "LinearAlgebra" }{TEXT -1 37 " package, solves the m atrix equation " }{XPPEDIT 18 0 "A*`.`*x=b" "6#/*(%\"AG\"\"\"%\".GF&% \"xGF&%\"bG" }{TEXT -1 17 ", for the vector " }{TEXT 290 1 "x" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x := LinearSolve(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'RTABLEG6$\"*!e#RZ\"-%'MATRIXG6#7%7##\"#`\"#S7##\"#xF07# #!#@\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can check this vector solution using matrix multiplication." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A.x;\nx := 'x':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"* OXNa\"-%'MATRIXG6#7%7#!\"%7#\"\")7#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The same problem could be given in matrix form by setting up the matrix " }{TEXT 293 1 "A" }{TEXT -1 12 " and vector " }{TEXT 294 1 "b" }{TEXT -1 11 " \"by hand\"." }} {PARA 0 "" 0 "" {TEXT -1 33 "We could use either the original " } {TEXT 291 1 "A" }{TEXT -1 5 " and " }{TEXT 292 1 "b" }{TEXT -1 25 " or the ones supplied by " }{TEXT 0 14 "GenerateMatrix" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "with(LinearAlgebra):\nA := Matrix([[2, -4, -1], [3, 1, -2], [3, -1, 1]]);\nb := Vector([-4,8,1]);\nLinearSolve(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*oWj[\"-%'MATRIXG6#7%7%\"\"#! \"%!\"\"7%\"\"$\"\"\"!\"#7%F2F0F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"bG-%'RTABLEG6$\"*o;Z`\"-%'MATRIXG6#7%7#!\"%7#\"\")7#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*+R#G:-%'MATRIXG6#7%7## \"#`\"#S7##\"#xF.7##!#@\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Here is another system " }{XPPEDIT 18 0 "A*`.`* x=b" "6#/*(%\"AG\"\"\"%\".GF&%\"xGF&%\"bG" }{TEXT -1 46 " described an d solved entirely in matrix form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "A := Matrix([[3,5,-2,6,1],[ 7,2,-8,3,6],[6,2,7,10,4],\n [7,-1,3,0,3],[8,5,1,-7,-4]]):\n b := Vector([3,-4,2,7,5]):\nA,b;\nx := LinearSolve(A,b);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$-%'RTABLEG6$\"*WV_]\"-%'MATRIXG6#7'7'\"\"$\"\"&! \"#\"\"'\"\"\"7'\"\"(\"\"#!\")F,F/7'F/F3F2\"#5\"\"%7'F2!\"\"F,\"\"!F,7 '\"\")F-F0!\"(!\"%-F$6$\"*/\"=7:-F(6#7'7#F,7#F>7#F37#F27#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'RTABLEG6$\"*;\"G3:-%'MATRIXG6#7'7## \"%F`\"$0%7##!%9')F07##!$I#\"#F7##\"$f$F77##!&14\"F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 5 "Check" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A.x;\nx := 'x':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6 $\"*S-Z]\"-%'MATRIXG6#7'7#\"\"$7#!\"%7#\"\"#7#\"\"(7#\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Calculations involving matric es with floating point entries" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 181 "Normally Maple performs arith metic operations involving floating point numbers using \"software\" b ase 10 arithmetic, with the number of digits used controlled by the gl obal variable " }{TEXT 0 6 "Digits" }{TEXT -1 394 ". Each basic arithm etic operation of addition, subtraction, multiplication or division is broken up into individual simple steps whereby the calculation is per formed just using the arithmetic operations as they apply to positive \+ integers with a fixed (small) number of digits, similar to the way in \+ which addition, subtraction, (long) multiplication and (long) division are performed \"by hand\"." }}{PARA 0 "" 0 "" {TEXT -1 18 "For exampl e, with " }{TEXT 0 6 "Digits" }{TEXT -1 24 " set to 10, the command " }{TEXT 0 12 "evalf(ln(2))" }{TEXT -1 34 " gives a 10 digit value for l n(2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Digits := 10:\nevalf(ln(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1=ZJp!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Using " }{TEXT 0 6 "evalhf" }{TEXT -1 6 " . . \+ " }{HYPERLNK 17 "evalhf" 2 "evalhf" "" }{TEXT -1 13 " instead of " } {TEXT 0 5 "evalf" }{TEXT -1 161 ", causes Maple to use the floating po int arithmetic unit in the processor chip, that is, Maple performs the evaluation using hardware floating point arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A := Ma trix([[Pi,sqrt(2)],[exp(-1),ln(2)]]);\nevalhf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*!31m9-%'MATRIXG6#7$7$%#PiG*$-%%sqr tG6#\"\"#\"\"\"7$-%$expG6#!\"\"-%#lnG6#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*_8][\"-%'MATRIXG6#7$7$$\"37$z*e`EfTJ!#<$ \"3:&4tBc8UT\"F.7$$\"3MBWr6WzyO!#=$\"3'GX*f0=ZJpF4" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 4 "Note" }{TEXT -1 50 ": B y default, matrix arithmetic for matrices with " }{TEXT 259 22 "floati ng point entries" }{TEXT -1 24 " is performed using the " }{TEXT 270 31 "hardware floating point numbers" }{TEXT -1 129 ", which will gener ally give matrix entries with between 15 and 18 decimal digits regardl ess of the number digits given initially." }}{PARA 0 "" 0 "" {TEXT -1 52 "This behaviour is controlled by the global variable " }{TEXT 0 17 "UseHardwareFloats" }{TEXT -1 34 ", which has the default value of " }{TEXT 271 4 "true" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "UseHardwareFloats;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalhf(ln(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 3'GX*f0=ZJp!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Let's change this to \"fa lse\", and work with 10 digit \"" }{TEXT 270 8 "software" }{TEXT -1 25 "\" floating point numbers." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "UseHardwareFloats := false; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2UseHardwareFloatsG%&falseG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "UseHardwareFloats;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Given a \+ 3 " }{TEXT 295 1 "x" }{TEXT -1 10 " 3 matrix " }{TEXT 273 1 "A" } {TEXT -1 38 " with floating point entries, we find " }{XPPEDIT 18 0 "B =``" "6#/%\"BG%!G" }{XPPEDIT 18 0 "A^2 = A*`.`*A" "6#/*$%\"AG\"\"#*(F% \"\"\"%\".GF(F%F(" }{TEXT -1 25 ", and the inverse matrix " }{XPPEDIT 18 0 "C = A^(-1)" "6#/%\"CG)%\"AG,$\"\"\"!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 28 "Then we compute the product " }{XPPEDIT 18 0 "B*`.`*C" "6#*(%\"BG\"\"\"%\".GF%%\"CGF%" }{TEXT -1 54 ", which i s approximately equal to the original matrix " }{TEXT 272 1 "A" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "A := Matrix([[9.408754603,-5.272378103,-14.9164 7546],\n[-11.205834550,12.53735484,-6.332354061],\n[6.789450618,-2.075 898853,5.533219832]]);\nB := A^2;\nC := A^(-1);\nB.C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*W'=-:-%'MATRIXG6#7%7%$\"+.Yv3 %*!\"*$!+.\"yBF&F0$!+Yvk\"\\\"!\")7%$!,]X$e?6F0$\"+%[NPD\"F5$!+hSNKjF0 7%$\"+=1X*y'F0$!+`))*e2#F0$\"+K)>K`&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\"*c'=Y:-%'MATRIXG6#7%7%$\"*kQJj%!\"($!+g#4VZ) !\")$!++.&\\*=F07%$!+mn<*)GF0$\"+'*)>TH#F0$\"+>zAs_F37%$\"+m(*4Z7F0$!+ +O#4L(F3$!+EDG^dF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG 6$\"*Gg@Z\"-%'MATRIXG6#7%7%$\"+WZEgT!#6$\"+%)fp\\WF0$\"+I+wI;!#57%$\"+ uPl19F0$\"+)=WX8\"F5$\"+(y1wn\"F57%$!+=J0xXF0$!+t'eM?\"F0$\"+yVbcVF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*Kh$*\\\"-%'MATRIXG6#7 %7%$\"+.Yv3%*!\"*$!+0\"yBF&F.$!+Xvk\"\\\"!\")7%$!+bMe?6F3$\"+'[NPD\"F3 $!+eSNKjF.7%$\"+;1X*y'F.$!+f))*e2#F.$\"+B)>K`&F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 5 "Notes" }{TEXT -1 2 ": " } }{PARA 15 "" 0 "" {TEXT -1 146 "If we use hardware floating point arit hmetic, then a few more digits are used. This means that rounding erro rs are reduced during the computation." }}{PARA 15 "" 0 "" {TEXT -1 51 "The final result can be rounded to 10 digits using " }{TEXT 0 5 "e valf" }{TEXT -1 42 ", which must be applied via the procedure " } {TEXT 0 3 "map" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "UseHardwareFloats := true; \nA := Matrix([[9.408754603,-5.272378103,-14.91647546],\n[-11.20583455 0,12.53735484,-6.332354061],\n[6.789450618,-2.075898853,5.533219832]]) ;\nB := A^2;\nC := A^(-1);\nB.C;\nmap(evalf,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2UseHardwareFloatsG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*%Gad:-%'MATRIXG6#7%7%$\"+.Yv3%*!\" *$!+.\"yBF&F0$!+Yvk\"\\\"!\")7%$!,]X$e?6F0$\"+%[NPD\"F5$!+hSNKjF07%$\" +=1X*y'F0$!+`))*e2#F0$\"+K)>K`&F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"BG-%'RTABLEG6$\"*ORZX\"-%'MATRIXG6#7%7%$\"3'4dXcjQJj%!#;$!3K5)4vD4V Z)F0$!3g`'>4I]\\*=!#:7%$!3W>IFmn<*)GF5$\"3)4_Li*)>TH#F5$\"3q?_];zAs_F0 7%$\"3;P5^m(*4Z7F5$!3-n)R/gB4L(F0$!3#Q4-*GDG^dF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6$\"*c9=Z\"-%'MATRIXG6#7%7%$\"3]F/1WZEg T!#>$\"3Yw[p&)fp\\WF0$\"3?oyEI+wI;!#=7%$\"3nCT'[x`mS\"F0$\"3Fi)yv=WX8 \"F5$\"3;z#[ry1wn\"F57%$!3_![Dr6`qd%F0$!3GB)=1neM?\"F0$\"3)\\]?3QalN%F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*Oq_[\"-%'MATRIXG6# 7%7%$\"3A+++.Yv3%*!#<$!3?,++.\"yBF&F.$!35+++Yvk\"\\\"!#;7%$!30+++bMe?6 F3$\"3!******R[NPD\"F3$!3U)****41aBL'F.7%$\"3;*****zh]%*y'F.$!3m)****H &))*e2#F.$\"3c)****>$)>K`&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTA BLEG6$\"*kK7^\"-%'MATRIXG6#7%7%$\"+.Yv3%*!\"*$!+.\"yBF&F.$!+Yvk\"\\\"! \")7%$!+bMe?6F3$\"+%[NPD\"F3$!+hSNKjF.7%$\"+=1X*y'F.$!+`))*e2#F.$\"+K) >K`&F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Constructi ng random vectors and matrices" }}{PARA 0 "" 0 "" {TEXT -1 15 "The pro cedures " }{TEXT 0 12 "RandomVector" }{TEXT -1 5 " and " }{TEXT 0 12 " RandomMatrix" }{TEXT -1 8 " in the " }{TEXT 0 13 "LinearAlgebra" } {TEXT -1 60 " package can be used to construct vectors and matrices wi th " }{TEXT 259 14 "random entries" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "with(Linear Algebra):\nA := RandomMatrix(3,2);\nB := RandomMatrix(3);\nb := Rand omVector[column](3);\nB.A;\nB.b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AG-%'RTABLEG6$\"(_n\"y-%'MATRIXG6#7%7$\"#q!#87$\"#s\"#U7$!#y!#&*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\"(sy!z-%'MATRIXG6#7 %7%!#>\"#n\"#C7%!#Z\"#&)!#'*7%\"#m\"#6!#!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6$\"(3hm(-%'MATRIXG6#7%7#\"\"$7#!#<7#!# W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(+M(z-%'MATRIXG6#7% 7$\"%A;\"$\"y7$\"&=.\"\"&,L\"7$\"&KC\"\"%a\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(on1)-%'MATRIXG6#7%7#!%_A7#\"%QE7#\"%rR" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The ty pe of random entries can be controlled by the option \"generator=a . . b\". " }}{PARA 0 "" 0 "" {TEXT -1 67 "The default is \"generator=-99 \+ . . 99\" which gives 2 digit integers." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "If the range delimiters a and b are \+ " }{TEXT 259 22 "floating point numbers" }{TEXT -1 56 ", the resulting matrix will have floating point entries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A := RandomMatrix(2,g enerator=-9.0..9.0);\nB := A^(-1);\nA.B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"(o$)4)-%'MATRIXG6#7$7$$\"37Db!Hy@i^$!#<$\"3# zfJW\"o`(\\#!#=7$$!3ww3&H&**p$\\\"F0$!3=i!3IRh!fGF3" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\")72uE-%'MATRIXG6#7$7$$\"3nTDh^e-A X!#=$\"3S,mW*e@-&RF07$$!3IQ7BPg]iB!#<$!3AR;U0;UhbF6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6$\")W$*)p#-%'MATRIXG6#7$7$$\"\"\"\"\"!$\"3 38.D\\gW?A!#L7$$!3ac^iCIA56F1F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "Notice that, in general, the entries in t he product A " }{TEXT 266 1 "." }{TEXT -1 75 " B are close, but not ex actly equal to, the entries of the identity matrix." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ta sks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "A = matrix([[3, -4, 5], [8, 0, -3], [5, 2, 1]]);" "6#/%\"AG-%'ma trixG6#7%7%\"\"$,$\"\"%!\"\"\"\"&7%\"\")\"\"!,$F*F-7%F.\"\"#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "B = matrix([[10, -6, -9], [6, -5, - 7], [-10, 9, 12]]);" "6#/%\"BG-%'matrixG6#7%7%\"#5,$\"\"'!\"\",$\"\"*F -7%F,,$\"\"&F-,$\"\"(F-7%,$F*F-F/\"#7" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Find" }}{PARA 0 "" 0 "" {TEXT -1 6 "(a) " }{XPPEDIT 18 0 "A + 4*B" "6#,&%\"AG\"\"\"*&\"\"%F%%\"BGF%F%" }{TEXT -1 15 ", \+ (b) " }{XPPEDIT 18 0 "B - 3*A" "6#,&%\"BG\"\"\"*&\"\"$F%%\"AGF%! \"\"" }{TEXT -1 16 ", (c) " }{XPPEDIT 18 0 "A*`.`*B;" "6#*(% \"AG\"\"\"%\".GF%%\"BGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "A^(-1);" "6# )%\"AG,$\"\"\"!\"\"" }{TEXT -1 14 ", (e) " }{XPPEDIT 18 0 "B^( -1);" "6#)%\"BG,$\"\"\"!\"\"" }{TEXT -1 16 ", (f) " } {XPPEDIT 18 0 "(A*`.`*B)^(-1);" "6#)*(%\"AG\"\"\"%\".GF&%\"BGF&,$F&!\" \"" }{TEXT -1 16 ", (g) " }{XPPEDIT 18 0 "B^(-1)*`.`*A^(-1); " "6#*()%\"BG,$\"\"\"!\"\"F'%\".GF')%\"AG,$F'F(F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 20 ": In order to enter " }{TEXT 260 1 "A" }{TEXT -1 5 " and \+ " }{TEXT 260 1 "B" }{TEXT -1 81 " on a command line, just select the a ppropriate matrix above, and use copy-paste." }}{PARA 0 "" 0 "" {TEXT 265 7 "Warning" }{TEXT -1 61 ": You will need to change the lower case m in \"matrix\" to an " }{TEXT 259 10 "upper case" }{TEXT -1 3 " M." }}{PARA 0 "" 0 "" {TEXT -1 49 "_______________________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 49 "_______________________________ __________________" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Answers for parts (d) to (f)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "A := M atrix([[3,-4,5],[8,0,-3],[5,2,1]]);\nB := Matrix([[10,-6,-9],[6,-5,-7] ,[-10,9,12]]);\nAinv := A^(-1);\nBinv := B^(-1);\nC := A.B;\nCinv := C ^(-1);\nBinv.Ainv;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG 6$\"()[-\")-%'MATRIXG6#7%7%\"\"$!\"%\"\"&7%\"\")\"\"!!\"$7%F0\"\"#\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\"(s+8)-%'MATR IXG6#7%7%\"#5!\"'!\"*7%\"\"'!\"&!\"(7%!#5\"\"*\"#7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%AinvG-%'RTABLEG6$\"(W9?)-%'MATRIXG6#7%7%#\"\"$\"#& *#\"\"(F0#\"\"'F07%#!#B\"$!>#!#6F0#\"#\\F87%#\"\")F0#!#8F0#\"#;F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%BinvG-%'RTABLEG6$\"(g*y\")-%'MATRIX G6#7%7%#\"\"\"\"\"##!\"$F0#!\"\"F07%#F4\"\"$\"\"&#\"\")F77%#F0F7!\"&#! \"(F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6$\")O(*fE-%' MATRIXG6#7%7%!#W\"#Z\"#h7%\"$5\"!#v!$3\"7%\"#_!#J!#Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%CinvG-%'RTABLEG6$\"(Wr3)-%'MATRIXG6#7%7%#\"#f\"$! Q#\"#`\"$!>#!$n\"F07%#!$B#\"$q&#!##*\"#&*#\"$z*F97%#\"#\\\"$9\"#\"#=\" #>#!$(=FB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")SShE-%'MAT RIXG6#7%7%#\"#f\"$!Q#\"#`\"$!>#!$n\"F.7%#!$B#\"$q&#!##*\"#&*#\"$z*F77% #\"#\\\"$9\"#\"#=\"#>#!$(=F@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "Notice that " }{XPPEDIT 18 0 "(A*B)^(-1) \+ = B^(-1)*A^(-1)" "6#/)*&%\"AG\"\"\"%\"BGF',$F'!\"\"*&)F(,$F'F*F')F&,$F 'F*F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 " " {TEXT -1 28 "(a) Solve the linear system " }{XPPEDIT 18 0 "A*`.`*x=b " "6#/*(%\"AG\"\"\"%\".GF&%\"xGF&%\"bG" }{TEXT -1 7 " where " } {XPPEDIT 18 0 "A = matrix([[4, 5, -5, -8, -2], [7, 2, -10, -1, -6], [6 , 2, 10, -10, 7], [-8, -1, -4, 0, 3], [8, -7, -3, 10, 5]]);" "6#/%\"AG -%'matrixG6#7'7'\"\"%\"\"&,$F+!\"\",$\"\")F-,$\"\"#F-7'\"\"(F1,$\"#5F- ,$\"\"\"F-,$\"\"'F-7'F9F1F5,$F5F-F37',$F/F-,$F7F-,$F*F-\"\"!\"\"$7'F/, $F3F-,$FAF-F5F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = matrix([[5], \+ [-4], [-7], [5], [7]]);" "6#/%\"bG-%'matrixG6#7'7#\"\"&7#,$\"\"%!\"\"7 #,$\"\"(F.7#F*7#F1" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ (b) Check your solution." }}{PARA 0 "" 0 "" {TEXT -1 49 "_____________ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 49 "_______________________________ __________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 4 " Let " }{XPPEDIT 18 0 "A = matrix([[9.408754603,-5.272378103,-14.916475 46],\n[-11.205834550,12.53735484,-6.332354061],\n[6.789450618,-2.07589 8853,5.533219832]])" "6#/%\"AG-%'matrixG6#7%7%-%&FloatG6$\"+.Yv3%*!\"* ,$-F+6$\"+.\"yBF&F.!\"\",$-F+6$\"+Yvk\"\\\"!\")F37%,$-F+6$\",]X$e?6F.F 3-F+6$\"+%[NPD\"F8,$-F+6$\"+hSNKjF.F37%-F+6$\"+=1X*y'F.,$-F+6$\"+`))*e 2#F.F3-F+6$\"+K)>K`&F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "(a) Find " }{XPPEDIT 18 0 "B = A^11" " 6#/%\"BG*$%\"AG\"#6" }{TEXT -1 30 ", which is the matrix product " } {XPPEDIT 18 0 "A*`.`*A*`.`*A*`.`*A*`.`*A*`.`*A*`.`*A*`.`*A*`.`*A*`.`*A *`.`*A" "6#*L%\"AG\"\"\"%\".GF%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%" }{TEXT -1 4 " of " }{TEXT 276 1 "A" }{TEXT -1 22 " with itself 11 times." }}{PARA 0 "" 0 "" {TEXT -1 9 "(b) Find " }{XPPEDIT 18 0 "C = A^(-10);" "6#/%\"CG)%\"AG,$\"#5! \"\"" }{TEXT -1 30 ", which is the matrix product " }}{PARA 258 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "A^(-1)*`.`*A^(-1)*`.`*A^(-1)*`.`*A^(- 1)*`.`*A^(-1)*`.`*A^(-1)*`.`*A^(-1)*`.`*A^(-1)*`.`*A^(-1)*`.`*A^(-1); " "6#*H)%\"AG,$\"\"\"!\"\"F'%\".GF')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'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "of the inverse of " }{TEXT 278 1 "A" }{TEXT -1 22 " with itself 10 times." }}{PARA 0 "" 0 "" {TEXT -1 24 "(c) Compute the product " } {XPPEDIT 18 0 "B*`.`*C" "6#*(%\"BG\"\"\"%\".GF%%\"CGF%" }{TEXT -1 41 " , which should be approximately equal to " }{TEXT 277 1 "A" }{TEXT -1 38 ", except for possible rounding errors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Perform the following computati on in " }{TEXT 257 8 "two ways" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 18 " (i) Use 10 digit " }{TEXT 270 34 "software floating poin t arithmetic" }{TEXT -1 18 " via the commands " }{TEXT 260 12 "Digits \+ := 10" }{TEXT -1 5 " and " }{TEXT 260 24 "UseHardwareFloats:=false" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 " (ii) Use " }{TEXT 270 34 "hardware floating point arithmetic" }{TEXT -1 17 " via the command " }{TEXT 260 24 "UseHardwareFloats:=false" }{TEXT -1 42 ", and round \+ the result to 10 digits using " }{TEXT 0 9 "map/evalf" }{TEXT -1 38 " \+ as in the last but one section above." }}{PARA 0 "" 0 "" {TEXT -1 49 " _________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 49 "____________ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 18 "(a) Construct a 4 " }{TEXT 279 1 "x" } {TEXT -1 10 " 4 matrix " }{TEXT 280 1 "A" }{TEXT -1 29 ", and a 4-dime nsional vector " }{TEXT 281 1 "b" }{TEXT -1 53 ", each with random 10 \+ digit entries between -5 and 5." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) Solve the linear system " } {XPPEDIT 18 0 "A*`.`*x=b" "6#/*(%\"AG\"\"\"%\".GF&%\"xGF&%\"bG" } {TEXT -1 16 " for the matrix " }{TEXT 282 1 "A" }{TEXT -1 12 " and vec tor " }{TEXT 283 1 "b" }{TEXT -1 20 " constructed in (a)." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(c) Check your sol ution." }}{PARA 0 "" 0 "" {TEXT -1 49 "_______________________________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 49 "_______________________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 89 "Check the left- hand distributive property of matrix multiplication over matrix additi on: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*`.`*(B+C)=A *`.`*B+A*`.`*C" "6#/*(%\"AG\"\"\"%\".GF&,&%\"BGF&%\"CGF&F&,&*(F%F&F'F& F)F&F&*(F%F&F'F&F*F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 " using 3 " }{TEXT 284 1 "x" }{TEXT -1 12 " 3 matrices " }{TEXT 285 1 "A " }{TEXT -1 2 ", " }{TEXT 286 1 "B" }{TEXT -1 5 " and " }{TEXT 287 1 " C" }{TEXT -1 37 " with random 2 digit integer entries." }}{PARA 0 "" 0 "" {TEXT -1 49 "_________________________________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 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