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All statements a re to appear on the worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 161 "Annotate your conclusions so that the re ader does not have to interpolate the relationships between the stated exam questions and the printed Maple computations." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Turn in a printed versi on of worksheet at the end of the testing period today. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 373 "Should there be p roblems on the exam for which you do not find the answers during the t esting period today, you may submit an electronic version of the works heet to me (through the WebCT e-mail interface or to pearce@math.ttu.e du) by 4:00 pm tomorrow and I will grade those problems at half credit and add their scores to the scores on the printed version of the work sheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Please provide to me: Last 3 Digits of Your Social Security Numb er (S) _____" }}{PARA 0 "" 0 "" {TEXT -1 113 " Number of the Month of Your Birthdat e (M) _____" }}{PARA 0 "" 0 "" {TEXT -1 106 " Number of the Day-of-the-Mon th of Your Birthdate (D) _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "1. (8 points) An " }{TEXT 289 10 "arithmetic" }{TEXT -1 147 " sequence is one in which the diffe rence between consective terms is always the same. In such a sequence there is an identifiable first term, say " }{TEXT 290 1 "a" }{TEXT -1 37 ", and a fixed common difference, say " }{TEXT 291 1 "d" }{TEXT -1 41 ". Then, the sequence takes on the form \{" }{TEXT 292 10 "a, a +d, a+" }{TEXT -1 1 "2" }{TEXT 295 5 "d, a+" }{TEXT -1 1 "3" }{TEXT 296 5 "d, a+" }{TEXT -1 1 "4" }{TEXT 297 9 "d, . . ." }{TEXT -1 94 " \}. Use Maple to construct the first thirty terms of the arithmetic s equence whose first term " }{TEXT 293 1 "a" }{TEXT -1 74 " is the numb er of month of your birthdate (M) and whose common difference " } {TEXT 294 1 "d" }{TEXT -1 136 " is the number of the day-of-the-month \+ of your birthdate (D) if the last 3 digits of your social security num ber (S) is even, otherwise " }{TEXT 298 1 "d" }{TEXT -1 8 " is the " } {TEXT 299 8 "negative" }{TEXT -1 128 " of the number of the day-of-the -month of your birthdate (-D) if if the last 3 digits of your social s ecurity number (S) is odd." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(3 + k*27,k=0..29);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6@\"\"$\"#I\"#d\"#%)\"$6\"\"$Q\"\"$l\"\"$#>\"$># \"$Y#\"$t#\"$+$\"$F$\"$a$\"$\"Q\"$3%\"$N%\"$i%\"$*[\"$;&\"$V&\"$q&\"$( f\"$C'\"$^'\"$y'\"$0(\"$K(\"$f(\"$'y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "2. (8 points) A " }{TEXT 300 9 "geometric" }{TEXT -1 153 " sequence is one in which the ratio between consecutive terms is \+ always the same. In such a sequence there is an identifiable non-zer o first term, say " }{TEXT 301 1 "a" }{TEXT -1 32 ", and a fixed commo n ratio, say " }{TEXT 302 1 "r" }{TEXT -1 48 ". Then, the sequence ta kes on the form \{a, ar, " }{XPPEDIT 18 0 "ar^2;" "6#*$%#arG\"\"#" } {TEXT -1 3 " , " }{XPPEDIT 18 0 "ar^3;" "6#*$%#arG\"\"$" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "ar^4;" "6#*$%#arG\"\"%" }{TEXT -1 101 ", . . . \}. Use Maple to construct the first thirty terms of the geometric seque nce whose first term " }{TEXT 303 1 "a" }{TEXT -1 89 " is the last thr ee digits of your social security number (S) and and whose common rati on " }{TEXT 304 1 "r" }{TEXT -1 120 " is the ratio of your number the \+ month of your birthdate to the number of the day-of-the-month of your \+ birthdate (M/D). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq(494*(3/27)^k,k=0..29);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6@\"$%\\#F#\"\"*#F#\"#\")#F#\"$H(#F#\"%hl#F#\"&\\!f#F# \"'T9`#F#\"(pHy%#F#\")@n/V#F#\"**[?uQ#F#\"+,Wy'[$#F#\",4'f5QJ#F#\"-\"[ O&HCG#F#\".H$Ge'=a##F#\"/h\\X#zwG##F#\"0\\Y4K6*e?#F#\"1T=&))=?I&=#F#\" 2plm*p\"=xm\"#F#\"3@\"**pHNY4]\"#F#\"4*3#*Hn<<&3N\"#F#\"5,)Gp0famd@\"# F#\"64#fB^J\"*)*=%4\"#F#\"6\")GB6O=-4x%)*#F#\"7Hf4,Dl>\"QH'))#F#\"8hL' )4D(o2Vkwz#F#\"9\\-x)e_=p()z*yr#F#\":TA$*)HtmA*)=3hk#F#\";p,R!pf+/.qt \\\"e#F#\"<@:^8s`gt-LwMB&#F#\"=*o.;#\\$[CYspG,r%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 19 "3. (8 points) The " }{TEXT 256 9 "Fibonacci" } {TEXT -1 333 " sequence is neither arithmetic for geometric. It is a \+ recursive sequence, whose first two terms are both 1 and for which the reafter each successive term is the sum of the two preceding terms. T hus, the sequence takes on the form \{1, 1, 2, 3, 5, 8, . . . \}. Use Maple to construct the first thirty terms of the Fibonacci sequence. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f[1] := 1; f[2] := 1;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\"F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"fG6#\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for k from 3 to 30 do f[k] := f[k-1] + f[k-2] od:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(f[k],k=1..30);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6@\"\"\"F#\"\"#\"\"$\"\"&\"\")\"#8\"#@\" #M\"#b\"#*)\"$W\"\"$L#\"$x$\"$5'\"$()*\"%(f\"\"%%e#\"%\"=%\"%ln\"&Y4\" \"&6x\"\"&d'G\"&oj%\"&D](\"'$R@\"\"'=k>\"'6yJ\"'HU^\"'S?$)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 204 "4. (16 points) Solve the following li near system by using elementary row operations and back substitution o n its augmented matrix. Clearly identify the solution set for this sy stem of linear equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,**&\"\"#\"\"\"&%\"xG6#F'F'F'*&F&F'&F)6#F&F'F' *&\"\"%F'&F)6#\"\"$F'F'*&\"\"&F'&F)6#F/F'F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"xG6#\"\"\"F(*&\"\"$F(&F&6#\"\"#F(F(*&F-F(&F&6#F* F(!\"\"*&\"\"&F(&F&6#\"\"%F(F(*&\"\"'F(&F&6#F3F(F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&\"\"%\"\"\"&%\"xG6#F'F'F'*&\"\")F'&F)6#\"\"#F'F '*&F&F'&F)6#\"\"$F'F'*&\"\"&F'&F)6#F&F'F'*&\"\"'F'&F)6#F5F'F'\"\"!" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiago nalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*WronskianG%'ad dcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG%&bas isG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG%'cold imG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG%*cross prodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(do tprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermat rixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+forward subG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG% )hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*inde xfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG% )jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'matad dG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*nor malizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatr ixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG% %rrefG%*scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*submatrixG%*s ubvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&trace G%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A := matrix( [ [2,2,4,5 ,0,-2], [1,3,-2,5,6,3], [4,8,4,5,6,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"#F*\"\"%\"\"&\"\"!!\"#7(\"\"\"\"\"$F.F ,\"\"'F17(F+\"\")F+F,F2F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A1 := swaprow(A,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%' matrixG6#7%7(\"\"\"\"\"$!\"#\"\"&\"\"'F+7(\"\"#F0\"\"%F-\"\"!F,7(F1\" \")F1F-F.F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A2 := addrow (A1,1,2,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'matrixG6#7%7( \"\"\"\"\"$!\"#\"\"&\"\"'F+7(\"\"!!\"%\"\")!\"&!#7!\")7(\"\"%F2F7F-F.F 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A3 := addrow(A2,1,3,-4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G-%'matrixG6#7%7(\"\"\"\"\" $!\"#\"\"&\"\"'F+7(\"\"!!\"%\"\")!\"&!#7!\")7(F0F1\"#7!#:!#=F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A4 := mulrow(A3,2,-1/4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G-%'matrixG6#7%7(\"\"\"\"\"$!\"# \"\"&\"\"'F+7(\"\"!F*F,#F-\"\"%F+\"\"#7(F0!\"%\"#7!#:!#=!#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A5 := addrow(A4,2,3,4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G-%'matrixG6#7%7(\"\"\"\"\"$!\"# \"\"&\"\"'F+7(\"\"!F*F,#F-\"\"%F+\"\"#7(F0F0F2!#5!\"'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A6 := mulrow(A5,3,1/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A6G-%'matrixG6#7%7(\"\"\"\"\"$!\"#\"\"&\" \"'F+7(\"\"!F*F,#F-\"\"%F+\"\"#7(F0F0F*#!\"&F3#!\"$F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Sol :=backsub(A6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SolG-%'vectorG6#7',(\"\"\"F**&#\"#X\"\"%F*&%#_t G6#\"\"#F*!\"\"*&\"\"$F*&F06#F*F*F3,$*&#\"#:F.F*F/F*F*,(F*F3*&#\"\"&F2 F*F/F*F**&#F5F2F*F6F*F*F/F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "for i from 1 to 5 do x[i] = Sol[i] od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\",(F'F'*&#\"#X\"\"%F'&%#_tG6#\"\"#F'!\"\" *&\"\"$F'&F.F&F'F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#,$ *&#\"#:\"\"%\"\"\"&%#_tGF&F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"xG6#\"\"$,(\"\"\"!\"\"*&#\"\"&\"\"#F)&%#_tG6#F.F)F)*&#F'F.F)&F06#F)F )F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"%&%#_tG6#\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"&&%#_tG6#\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "5. (12 points) Find all value s of a and b for which the following system of equations has" }}{PARA 0 "" 0 "" {TEXT -1 31 " i. a unique solution" }}{PARA 0 "" 0 "" {TEXT -1 45 " ii. an infinite number of solutions" }} {PARA 0 "" 0 "" {TEXT -1 25 " iii. no solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG\"\"\"%\"yGF&%\"zGF&%\"bG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"#\"\"\"%\"xGF'F'*&F&F'%\"yGF'F'*&%\"aGF'% \"zGF'F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"#\"\"\"%\"xGF'! \"\"*&F&F'%\"yGF'F)*&%\"bGF'%\"zGF'F'*$)F-F&F'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "restart: with(linalg):\nCM := matrix([ [1,1,1], [2,2,a], [-2,-2,b] ]);\nM := matrix([ [1,1,1,b],[2,2,a,2], [-2,-2,b,b^ 2] ]);" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names no rm and trace have been redefined and unprotected\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#CMG-%'matrixG6#7%7%\"\"\"F*F*7%\"\"#F,%\"aG7%!\"#F /%\"bG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7&\"\"\" F*F*%\"bG7&\"\"#F-%\"aGF-7&!\"#F0F+*$)F+F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(CM);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 314 61 "No case of a and b for which there is a unique solution set.\n" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "M1 := addrow(M,1,2,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7%7&\"\"\"F*F*%\"bG7&\"\"!F-,&%\"aGF* \"\"#!\"\",&*&F0F*F+F*F1F0F*7&!\"#F5F+*$)F+F0F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "M2 := addrow(M1,1,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'matrixG6#7%7&\"\"\"F*F*%\"bG7&\"\"!F-,&%\"aGF* \"\"#!\"\",&*&F0F*F+F*F1F0F*7&F-F-,&F+F*F0F*,&*&F0F*F+F*F**$)F+F0F*F* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 315 91 "If a = 2 and b<>1, then no \+ solutions.\nIf a = 2 and b = 1, then infinitely many solutions.\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 316 53 "If a <> 2 and b = -2, the n infinitely many solutions\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M3 := addrow(M2,2,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3 G-%'matrixG6#7%7&\"\"\"F*F*%\"bG7&\"\"!F-,&%\"aGF*\"\"#!\"\",&*&F0F*F+ F*F1F0F*7&F-F-,&F/F*F+F*,&F0F**$)F+F0F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 305 28 "If a+b=0, then no solutions." }}{PARA 0 " " 0 "" {TEXT 313 28 "\nSuppose a <> 2 and b <> -2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M4 := mulrow(M3,2,1/(a-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M4G-%'matrixG6#7%7&\"\"\"F*F*%\"bG7&\"\"!F-F**& ,&%\"aGF*\"\"#!\"\"F2,&*&F1F*F+F*F2F1F*F*7&F-F-,&F0F*F+F*,&F1F**$)F+F1 F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "M5 := addrow(M4,2,3 ,-(a+b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M5G-%'matrixG6#7%7&\" \"\"F*F*%\"bG7&\"\"!F-F**&,&%\"aGF*\"\"#!\"\"F2,&*&F1F*F+F*F2F1F*F*7&F -F-F-,(*(,&F0F2F+F2F*F/F2F3F*F*F1F**$)F+F1F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "map(simplify,M5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&\"\"\"F(F(%\"bG7&\"\"!F+F(,$*(\"\"#F(,& F)F(F(!\"\"F(,&%\"aGF(F.F0F0F07&F+F+F+*&,**(F.F(F2F(F)F(F(*&F.F(F)F(F0 \"\"%F0*&)F)F.F(F2F(F(F(F1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "solve((2*a*b-2*b-4+b^2*a)/(a-2)=0,a);\nsolve((2*a*b-2*b-4+b^2*a) /(a-2)=0,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%\"bG! \"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"#,$*&\"\"#\"\"\"%\"aG!\" \"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 318 60 "Under the above suppostion that a <> 2 and b <> -2, we have " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 30 "If ab <> 2, then no so lutions." }}{PARA 0 "" 0 "" {TEXT 312 42 "If ab = 2, then infinitely m any solutions." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "6. (18 points) Using M aple, use three different ways to solve the following system of linear equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG!\"\"%\"yGF&*&\"\"#\"\"\"%\"zGF*F*F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"$\"\"\"%\"xGF'!\"\"*&\"\"#F'%\"yGF' F'%\"zGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"xG\"\"\"%\"yG!\" \"%\"zGF&F&" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "A := matrix([ [-1,-1,2],[-3,2,-1],[1,-1,1] ]);\nB := matrix(3,1,[-1,-1,1]);\nC := a ugment(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7% !\"\"F*\"\"#7%!\"$F+F*7%\"\"\"F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"BG-%'matrixG6#7%7#!\"\"F)7#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7%7&!\"\"F*\"\"#F*7&!\"$F+F*F*7&\"\"\"F*F/F/" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 35 "Method #1 Gauss-J ordan elimination" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(C) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7&\"\"\"\"\"!F)\"\" $7&F)F(F)\"\"'7&F)F)F(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x=3,y=6,z=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"xG\"\"$/%\"yG \"\"'/%\"zG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 46 "Method #2 Multiply matrix equation AX = B by " }{XPPEDIT 268 0 "A ^\{-1\};" "6#)%\"AG<#,$\"\"\"!\"\"" }{TEXT 267 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalm(inverse(A)&*B);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7%7#\"\"$7#\"\"'7#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x=3,y=6,z=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"xG\"\"$/%\"yG\"\"'/%\"zG\"\"%" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 269 27 "Method #3 Use solve command" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{-x-y+2*z = -1,-3*x+2 *y-z = -1,x-y+z = 1\},\{x,y,z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#< %/%\"xG\"\"$/%\"yG\"\"'/%\"zG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 311 30 "Method #4 Use Linsolve command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#\"\"$7#\"\"'7#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x=3,y=6,z=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%\"xG\"\"$/%\"yG\"\"'/%\"zG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 23 "Method #5 Cramer's Rule" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "Ax := matrix([ [-1,-1,2],[-1,2,-1],[1,-1 ,1] ]);\nAy := matrix([ [-1,-1,2],[-3,-1,-1],[1,1,1] ]);\nAz := matrix ([ [-1,-1,-1],[-3,2,-1],[1,-1,1] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#AxG-%'matrixG6#7%7%!\"\"F*\"\"#7%F*F+F*7%\"\"\"F*F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#AyG-%'matrixG6#7%7%!\"\"F*\"\"#7%!\"$F*F*7%\" \"\"F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AzG-%'matrixG6#7%7%!\" \"F*F*7%!\"$\"\"#F*7%\"\"\"F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x=det(Ax)/det(A);y=det(Ay)/det(A);z=det(Az)/det(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"zG\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 69 " Note: Only one of the three methods may use an augmented m atrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "7. (30 points) Consider the matrix " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "A = matrix([[0, -1, -2, -3, -4, -5], [1, 0, -1, -2, -3, -4], [2, 1, 0, -1, -2, -3]]);" "6#/%\"AG-%'matrixG6#7%7(\"\"!,$\"\"\"!\"\",$\" \"#F-,$\"\"$F-,$\"\"%F-,$\"\"&F-7(F,F*,$F,F-,$F/F-,$F1F-,$F3F-7(F/F,F* ,$F,F-,$F/F-,$F1F-" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 " Let " }{XPPEDIT 18 0 "A^T;" "6#)%\"AG %\"TG" }{TEXT -1 25 " denote the transpose of " }{TEXT 258 2 "A." } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " i. Find an orthonormal basis (call it " }{TEXT 274 3 "BNS" }{TEXT -1 24 ") for the nullspace of " }{TEXT 257 1 "A" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " ii. Show that the column space of " }{XPPEDIT 18 0 "A^T;" "6#)%\"AG%\"TG" }{TEXT -1 34 " is the same as the row spac e of " }{TEXT 259 2 "A." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 46 " iii. Find an othonormal basis (call it " } {TEXT 273 3 "BCS" }{TEXT -1 29 " ) for the column space of " } {XPPEDIT 18 0 "A^T;" "6#)%\"AG%\"TG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " iv. Show that \+ " }{TEXT 275 3 "BNS" }{TEXT -1 19 " is orthogonal to " }{TEXT 277 3 " BCS" }{TEXT -1 22 ", i.e., every vector " }{TEXT 260 1 "v" }{TEXT -1 4 " in " }{TEXT 276 3 "BNS" }{TEXT -1 31 " is orthogonal to every vect or " }{TEXT 261 1 "w" }{TEXT -1 5 " in " }{TEXT 278 3 "BCS" }{TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 " " 0 "" {TEXT -1 14 " v. Let " }{TEXT 262 1 "B" }{TEXT -1 18 " be the union of " }{TEXT 279 3 "BNS" }{TEXT -1 6 " and " }{TEXT 280 3 "BCS" }{TEXT -1 21 ". Determine whether " }{TEXT 263 1 "B" }{TEXT -1 17 " is a basis for " }{XPPEDIT 18 0 "R^6;" "6#*$%\"RG\"\"'" }{TEXT -1 6 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "A := matrix( [[0, -1, -2, -3, -4, -5], [1, 0, -1, -2, -3, -4], [2, 1, 0, -1, -2, -3 ]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7(\"\"!!\" \"!\"#!\"$!\"%!\"&7(\"\"\"F*F+F,F-F.7(\"\"#F1F*F+F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "NS := nullspace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NSG<&-%'vectorG6#7(\"\"\"!\"#F*\"\"!F,F,-F'6#7( \"\"#!\"$F,F*F,F,-F'6#7(\"\"$!\"%F,F,F*F,-F'6#7(\"\"%!\"&F,F,F,F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "GNS := GramSchmidt(NS);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GNSG<&7(\"\"\"!\"#F'\"\"!F)F)7(#\" \"#\"\"$#!\"\"F-#!\"%F-F'F)F)7(#F'F,F)#F/F,F/F'F)7(#F,\"\"&#F'\"#5#F/F 7F4#F1F7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "BNS := seq(sc alarmul(GNS[i],1/norm(GNS[i],frobenius)),i=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$BNSG6&-%'vectorG6#7(,$*&\"\"'!\"\"F,#\"\"\"\"\"#F/,$ *&\"\"$F-F,F.F-F*\"\"!F4F4-F'6#7(,$*&\"#:F-\"#IF.F/,$*&F;F-F;F.F-,$*(F 0F/F:F-F;F.F-,$*&\"#5F-F;F.F/F4F4-F'6#7(,$*&FBF-FBF.F/F4,$*&FBF-FBF.F- ,$*&\"\"&F-FBF.F-,$*&FLF-FBF.F/F4-F'6#7(,$*(F0F/\"$0\"F-\"$5#F.F/,$*&F UF-FUF.F/,$*&FTF-FUF.F-,$*&\"#UF-FUF.F-,$*(\"\"%F/FTF-FUF.F-,$*&\"#@F- FUF.F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "AT := transpose(A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ATG-%'matrixG6#7(7%\"\"!\"\" \"\"\"#7%!\"\"F*F+7%!\"#F.F*7%!\"$F0F.7%!\"%F2F07%!\"&F4F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RS := rowspace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#RSG<$-%'vectorG6#7(\"\"\"\"\"!!\"\"!\"#!\"$!\"% -F'6#7(F+F*\"\"#\"\"$\"\"%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "CS := colspace(AT);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CSG< $-%'vectorG6#7(\"\"\"\"\"!!\"\"!\"#!\"$!\"%-F'6#7(F+F*\"\"#\"\"$\"\"% \"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 43 "\nHence, rowspace of A and column space of " }{XPPEDIT 18 0 "A^T" "6#)%\"AG% \"TG" }{TEXT 272 16 " are the same.\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "GCS := GramSchmidt(CS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GCSG<$7(#\"#S\"#J\"\"\"#\"#AF)#\"#8F)#\"\"%F)#!\"&F)7(F*\"\"! !\"\"!\"#!\"$!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "BCS := seq(scalarmul(GCS[i],1/norm(GCS[i],frobenius)),i=1..2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$BCSG6$-%'vectorG6#7(,$*(\"\")\"\"\"\"$^'!\"\" \"%bK#F-\"\"#F-,$*&\"$0\"F/F0F1F-,$*(\"#AF-F0F/F0F1F-,$*(\"#8F-F0F/F0F 1F-,$*(\"\"%F-F0F/F0F1F-,$*&F.F/F0F1F/-F'6#7(,$*&\"#JF/FFF1F-\"\"!,$*& FFF/FFF1F/,$*(F2F-FFF/FFF1F/,$*(\"\"$F-FFF/FFF1F/,$*(F>F-FFF/FFF1F/" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "for i from 1 to 4 do\nfor j from 1 to 2 do\nprint(BNS[i],`dot`,BCS[j],`value is`,dotprod(BNS[i] ,BCS[j]) )\nod;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%'vectorG6#7( ,$*&\"\"'!\"\"F)#\"\"\"\"\"#F,,$*&\"\"$F*F)F+F*F'\"\"!F1F1%$dotG-F$6#7 (,$*(\"\")F,\"$^'F*\"%bKF+F,,$*&\"$0\"F*F:F+F,,$*(\"#AF,F:F*F:F+F,,$*( \"#8F,F:F*F:F+F,,$*(\"\"%F,F:F*F:F+F,,$*&F9F*F:F+F*%)value~isGF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6'-%'vectorG6#7(,$*&\"\"'!\"\"F)#\"\"\" \"\"#F,,$*&\"\"$F*F)F+F*F'\"\"!F1F1%$dotG-F$6#7(,$*&\"#JF*F8F+F,F1,$*& F8F*F8F+F*,$*(F-F,F8F*F8F+F*,$*(F0F,F8F*F8F+F*,$*(\"\"%F,F8F*F8F+F*%)v alue~isGF1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%'vectorG6#7(,$*&\"#:! \"\"\"#I#\"\"\"\"\"#F-,$*&F+F*F+F,F*,$*(F.F-F)F*F+F,F*,$*&\"#5F*F+F,F- \"\"!F6%$dotG-F$6#7(,$*(\"\")F-\"$^'F*\"%bKF,F-,$*&\"$0\"F*F?F,F-,$*( \"#AF-F?F*F?F,F-,$*(\"#8F-F?F*F?F,F-,$*(\"\"%F-F?F*F?F,F-,$*&F>F*F?F,F *%)value~isGF6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%'vectorG6#7(,$*&\" #:!\"\"\"#I#\"\"\"\"\"#F-,$*&F+F*F+F,F*,$*(F.F-F)F*F+F,F*,$*&\"#5F*F+F ,F-\"\"!F6%$dotG-F$6#7(,$*&\"#JF*F=F,F-F6,$*&F=F*F=F,F*,$*(F.F-F=F*F=F ,F*,$*(\"\"$F-F=F*F=F,F*,$*(\"\"%F-F=F*F=F,F*%)value~isGF6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%'vectorG6#7(,$*&\"#5!\"\"F)#\"\"\"\"\"#F, \"\"!,$*&F)F*F)F+F*,$*&\"\"&F*F)F+F*,$*&F3F*F)F+F,F.%$dotG-F$6#7(,$*( \"\")F,\"$^'F*\"%bKF+F,,$*&\"$0\"F*F>F+F,,$*(\"#AF,F>F*F>F+F,,$*(\"#8F ,F>F*F>F+F,,$*(\"\"%F,F>F*F>F+F,,$*&F=F*F>F+F*%)value~isGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%'vectorG6#7(,$*&\"#5!\"\"F)#\"\"\"\"\"#F, \"\"!,$*&F)F*F)F+F*,$*&\"\"&F*F)F+F*,$*&F3F*F)F+F,F.%$dotG-F$6#7(,$*& \"#JF*F " 0 "" {MPLTEXT 1 0 62 "MB := matrix( [ BNS[ 1],BNS[2],BNS[3],BNS[4],BCS[1],BCS[2] ] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MBG-%'matrixG6#7(7(,$*&\"\"'!\"\"F,#\"\"\"\"\"#F/,$* &\"\"$F-F,F.F-F*\"\"!F4F47(,$*&\"#:F-\"#IF.F/,$*&F9F-F9F.F-,$*(F0F/F8F -F9F.F-,$*&\"#5F-F9F.F/F4F47(,$*&F@F-F@F.F/F4,$*&F@F-F@F.F-,$*&\"\"&F- F@F.F-,$*&FHF-F@F.F/F47(,$*(F0F/\"$0\"F-\"$5#F.F/,$*&FOF-FOF.F/,$*&FNF -FOF.F-,$*&\"#UF-FOF.F-,$*(\"\"%F/FNF-FOF.F-,$*&\"#@F-FOF.F/7(,$*(\"\" )F/\"$^'F-\"%bKF.F/,$*&FNF-F\\oF.F/,$*(\"#AF/F\\oF-F\\oF.F/,$*(\"#8F/F \\oF-F\\oF.F/,$*(FYF/F\\oF-F\\oF.F/,$*&F[oF-F\\oF.F-7(,$*&\"#JF-F\\pF. F/F4,$*&F\\pF-F\\pF.F-,$*(F0F/F\\pF-F\\pF.F-,$*(F3F/F\\pF-F\\pF.F-,$*( FYF/F\\pF-F\\pF.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rank(M B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 281 119 "Since the rank of MB is 6, the rows of MB are linearly indepe ndent, which implies that the rows of MB form a basis for " }{XPPEDIT 18 0 "R^6" "6#*$%\"RG\"\"'" }{TEXT 282 3 ". " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "69 0 1" 60 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }