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"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 "Heading 1" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 257 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 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 37 "Basis and Dimension of a Vector Space" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 29 "Linear Algebra Maple Project." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 310 10 "Objective:" }}{PARA 0 "" 0 "" {TEXT -1 30 "To introduce the concepts of " }{TEXT 287 5 "basis" }{TEXT -1 5 " and " }{TEXT 288 9 "dimension" }{TEXT -1 20 " of a vector space. \+ " }{TEXT 289 5 "Maple" }{TEXT -1 56 " will carry out the details of t he Gauss-Jordan method." }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 311 15 "Solved ex ample:" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{TEXT 290 1 "A" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[-1, -1, -2, 3, 1], [-9, 5, -4, -1, -5], [7, -5, 2, 3, 5]]);" "6#-%'matrixG6#7%7',$\"\"\"!\"\",$F)F*,$\" \"#F*\"\"$F)7',$\"\"*F*\"\"&,$\"\"%F*,$F)F*,$F2F*7'\"\"(,$F2F*F-F.F2" }{TEXT -1 27 " . The solution set for " }{TEXT 291 1 "A" }{TEXT 256 1 "x" }{TEXT -1 3 " = " }{TEXT 257 1 "0" }{TEXT -1 102 " forms a \+ vector space. Find a basis of this vector space. What is the dimension of this vector space?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 55 "We will solve the syst em using the Gauss-Jordan method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee n redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A:=matrix([[-1, -1, -2, 3, 1], [-9, 5, -4, -1, -5], [7, -5, 2, 3 , 5]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'!\"\" F*!\"#\"\"$\"\"\"7'!\"*\"\"&!\"%F*!\"&7'\"\"(F2\"\"#F,F0" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gaus sjord(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7'\"\"\"\" \"!F(!\"\"F)7'F)F(F(!\"#F*7'F)F)F)F)F)" }}}{PARA 257 "" 1 "" {TEXT -1 43 "We see that the only leading variables are " }{XPPEDIT 19 1 "x[1]; " "6#&%\"xG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 19 1 "x[2];" "6#&% \"xG6#\"\"#" }{TEXT -1 10 ". Taking " }{XPPEDIT 19 1 "x[5];" "6#&%\"x G6#\"\"&" }{TEXT -1 4 " as " }{XPPEDIT 19 1 "t[1];" "6#&%\"tG6#\"\"\" " }{TEXT -1 2 ", " }{XPPEDIT 19 1 "x[4];" "6#&%\"xG6#\"\"%" }{TEXT -1 4 " as " }{XPPEDIT 19 1 "t[2];" "6#&%\"tG6#\"\"#" }{TEXT -1 7 ", and \+ " }{XPPEDIT 19 1 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 5 " as " } {XPPEDIT 19 1 "t[3];" "6#&%\"tG6#\"\"$" }{TEXT -1 55 " , we can writ e the first two rows as equations for " }{XPPEDIT 19 1 "x[1];" "6#&% \"xG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 19 1 "x[2];" "6#&%\"xG6#\" \"#" }{TEXT -1 35 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The solution" } {TEXT 280 1 " " }{TEXT -1 2 "is" }{TEXT 281 2 " " }{TEXT 292 1 "x" } {TEXT -1 4 " = (" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[4];" "6#&%\"xG6#\"\"%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[5];" "6 #&%\"xG6#\"\"&" }{TEXT -1 14 "), where " }{XPPEDIT 18 0 "x[1];" " 6#&%\"xG6#\"\"\"" }{TEXT -1 5 " = - " }{XPPEDIT 18 0 "t[3];" "6#&%\"tG 6#\"\"$" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "t[2];" "6#&%\"tG6#\"\"#" }} {PARA 0 "" 0 "" {TEXT -1 78 " \+ " }{XPPEDIT 18 0 "x[2];" "6#&%\"x G6#\"\"#" }{TEXT -1 4 " = -" }{XPPEDIT 18 0 "t[3];" "6#&%\"tG6#\"\"$" }{TEXT -1 3 " +2" }{XPPEDIT 18 0 "t[2];" "6#&%\"tG6#\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "t[1];" "6#&%\"tG6#\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ " }{XPPEDIT 18 0 "x[3] = t[3];" "6#/&%\"xG6#\"\"$ &%\"tG6#F'" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ " }{XPPEDIT 18 0 "x [4] = t[2];" "6#/&%\"xG6#\"\"%&%\"tG6#\"\"#" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ " }{XPPEDIT 18 0 "x[5] = t[1];" "6#/&%\"x G6#\"\"&&%\"tG6#\"\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Any solution vector" }{TEXT 282 1 " \+ " }{TEXT 293 1 "x" }{TEXT -1 33 " can therefore be represented as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 44 " x = " }{XPPEDIT 18 0 " t[1];" "6#&%\"tG6#\"\"\"" }{XPPEDIT 18 0 "matrix([[0], [1], [0], [0], \+ [1]])+t[2]*matrix([[1], [2], [0], [1], [0]])+t[3]*matrix([[-1], [-1], \+ [1], [0], [0]]);" "6#,(-%'matrixG6#7'7#\"\"!7#\"\"\"7#F)7#F)7#F+F+*&&% \"tG6#\"\"#F+-F%6#7'7#F+7#F37#F)7#F+7#F)F+F+*&&F16#\"\"$F+-F%6#7'7#,$F +!\"\"7#,$F+FE7#F+7#F)7#F)F+F+" }{TEXT -1 7 " = " }{XPPEDIT 18 0 " t[1];" "6#&%\"tG6#\"\"\"" }{TEXT 260 1 "u" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "t[2];" "6#&%\"tG6#\"\"#" }{TEXT 261 1 "v" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "t[3];" "6#&%\"tG6#\"\"$" }{TEXT 262 1 "w" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 51 "which means that the solution \+ space is spanned by \{" }{TEXT 271 1 "u" }{TEXT -1 2 ", " }{TEXT 272 1 "v" }{TEXT -1 2 ", " }{TEXT 276 1 "w" }{TEXT -1 3 "\}. " }{TEXT 277 0 "" }{TEXT -1 97 "To see whether or not the vectors form a basis for \+ the solution space we need to find out whether" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 273 1 "u" }{TEXT -1 2 ", " }{TEXT 274 1 "v" } {TEXT -1 2 ", " }{TEXT 275 1 "w" }{TEXT -1 26 " are linearly independe nt." }}{PARA 0 "" 0 "" {TEXT -1 12 "The vectors " }{TEXT 256 1 "u" } {TEXT -1 2 ", " }{TEXT 257 1 "v" }{TEXT -1 2 ", " }{TEXT 258 3 "and" } {TEXT 309 3 " w" }{TEXT 283 1 " " }{TEXT -1 54 "are linearly independ ent if and only if the equation " }{XPPEDIT 18 0 "k[1];" "6#&%\"kG6# \"\"\"" }{TEXT 263 1 "u" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "k[2];" "6#& %\"kG6#\"\"#" }{TEXT -1 1 " " }{TEXT 265 1 "v" }{TEXT -1 4 " + " } {TEXT 264 0 "" }{XPPEDIT 18 0 "k[3];" "6#&%\"kG6#\"\"$" }{TEXT -1 1 " \+ " }{TEXT 266 1 "w" }{TEXT -1 4 " = " }{TEXT 267 1 "0" }{TEXT 294 1 " \+ " }{TEXT -1 32 " has the trivial solution only." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 "B:=matrix(5,3,[0,1,-1,1,2,-1,0,0,1,0,1,0,1,0,0 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7'7%\"\"!\"\" \"!\"\"7%F+\"\"#F,7%F*F*F+7%F*F+F*7%F+F*F*" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gaussjord(B);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7%\"\"\"\"\"!F)7%F)F(F)7 %F)F)F(7%F)F)F)F," }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 18 "The solution is " }{XPPEDIT 18 0 "k[1];" "6#&%\"kG6# \"\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "k[2];" "6#&%\"kG6#\"\"#" } {TEXT -1 5 " = " }{XPPEDIT 18 0 "k[3];" "6#&%\"kG6#\"\"$" }{TEXT -1 20 " = 0. The vectors " }{TEXT 268 1 "u" }{TEXT -1 2 ", " }{TEXT 269 1 "v" }{TEXT -1 7 ", and " }{TEXT 270 1 "w" }{TEXT 295 1 " " } {TEXT -1 27 " are linearly independent. " }}{PARA 0 "" 0 "" {TEXT -1 9 "Vectors \{" }{TEXT 278 1 " " }{TEXT 296 1 "u" }{TEXT 297 2 ", " } {TEXT 298 1 "v" }{TEXT 299 2 ", " }{TEXT 300 1 "w" }{TEXT 301 1 " " } {TEXT -1 52 "\} form a base. The dimension of the space is three." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 75 "________ ___________________________________________________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" } {TEXT 256 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 284 4 "Use " }{TEXT 285 5 "Maple" }{TEXT 286 31 " to solve systems of equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Problem 1." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Find a basis for " } {TEXT 302 1 "S" }{TEXT -1 8 " = span\{" }{TEXT 256 10 "r, u, v, w" } {TEXT -1 9 "\}, where " }{TEXT 257 1 "r" }{TEXT -1 20 " = (1, 3, 2, - 5), " }{TEXT 258 1 "u" }{TEXT -1 19 " = (0, 1, 5, -3), " }{TEXT 259 1 "v" }{TEXT -1 24 " = (4, 1, 1, -1), and " }{TEXT 260 1 "w" }{TEXT -1 44 " = (-2, 5, 3, -9). What is the dimension of " }{TEXT 303 1 "S" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Problem 2." }} {PARA 0 "" 0 "" {TEXT -1 99 "Find a basis for the solution set of 2x \+ - y + 4z = 0. What is the dimension of the solution space?" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 " " 0 "" {TEXT -1 10 "Problem 3." }}{PARA 0 "" 0 "" {TEXT -1 67 "Conside r the solution set of the system of equations 2" }{TEXT 304 1 "x" }{TEXT -1 3 " - " }{TEXT 306 1 "y" }{TEXT -1 4 " + 4" } {TEXT 307 1 "z" }{TEXT -1 4 " = 0" }}{PARA 0 "" 0 "" {TEXT -1 93 " \+ \+ " }{TEXT 305 1 "x" }{TEXT -1 9 " + 3" }{TEXT 308 1 "z" }{TEXT -1 6 " = 0 ." }}{PARA 0 "" 0 "" {TEXT -1 43 "a) Show \+ that this set forms a subspace of " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG \"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 40 "b) Find the base for the solution space." }}{PARA 0 "" 0 "" {TEXT -1 46 "c) What is th e dimension of this vector space?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 73 "_________________________________________ ________________________________" }}{PARA 0 "" 0 "" {TEXT -1 177 "MSIP Grant #P120A80089-98: \"Three Urban Calculus Reform Programs: Adoptin g the Best,\" 1998-2001; MSEIP Grant #P120AA010031: \"Four Colleges: \+ Calculus + Enhancements\", 2001-2004 " }}}{MARK "31 0" 177 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }