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Click on a [-] sign to collapse a section. To do these exercises y ou will have to insert execution groups. That can be done by clicking \+ on the toolbar icon that looks like \"[>\". It can also be done via th e Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 20 "Student Name and ID:" }} {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 8 "Keyword s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Cli cking 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 "augment" 2 "linalg[augment]" "" } {TEXT -1 1 "," }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "ditt o" 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 "lina lg" "" }{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 "matrix" 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 "restart" 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 "scalarmul" "" }{TEXT -1 1 "," }} {PARA 15 "" 0 "" {TEXT -1 3 " " }{HYPERLNK 17 "swaprow" 2 "linalg[sw aprow]" "" }{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 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Exercise 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 26 "Verify that the vectors " }{XPPEDIT 19 1 "M ATRIX([ [2] ,[-1],[1],[1],[3],[4] ] ),MATRIX([ [1] ,[1],[1],[1],[2],[0 ] ] ),MATRIX([ [2] ,[1],[1],[1],[1],[1] ] ),MATRIX([ [2] ,[0],[1],[-1] ,[1],[-1] ] )" "6&-%'MATRIXG6#7(7#\"\"#7#,$\"\"\"!\"\"7#F+7#F+7#\"\"$7 #\"\"%-F$6#7(7#F+7#F+7#F+7#F+7#F(7#\"\"!-F$6#7(7#F(7#F+7#F+7#F+7#F+7#F +-F$6#7(7#F(7#F<7#F+7#,$F+F,7#F+7#,$F+F," }{TEXT -1 29 " are linear ly independent." }}{PARA 0 "" 0 "" {TEXT -1 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 13 "Exercise 2 " }}{PARA 271 "" 0 "" {TEXT -1 6 "Let " }}{PARA 271 "" 0 "" {TEXT -1 1 " " }}{PARA 271 "" 0 "" {TEXT -1 24 " " }{XPPEDIT 19 1 "\{e[1]=[1,0,0,0 ,0,0],e[2]=[0,1,0,0,0,0],`...`,e[6]=[0,0,0,0,0,1]\}" "<&/&%\"eG6#\"\" \"7(F'\"\"!F)F)F)F)/&F%6#\"\"#7(F)F'F)F)F)F)%$...G/&F%6#\"\"'7(F)F)F)F )F)F'" }{TEXT -1 228 " \n\nbe the standard basis of the vector space V of real 6-tuples.\n\nApply the Replacement Theorem using the four vectors of Exercise 1 as the set of linearly independent vectors and \+ the standard basis as the spanning set of V." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 260 "" 0 "" {TEXT -1 11 "Exercise 3 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 39 "Calculate a basis for \+ the subspace of " }{XPPEDIT 19 1 "R^8" "*$%\"RG\"\")" }{TEXT 262 21 " that is spanned by" }{TEXT -1 8 " \n\n " }{XPPEDIT 263 1 "\{ V ECTOR([0, 0, 1, 1, 1, 1, 1, 1]), VECTOR([1, 0, 1, 0, 3, -2, 5, -8]), V ECTOR([2, 3, 0, -4, 3, 3, 8, 6]), VECTOR([-1, 2, 0, 0, 5, 0, 11, -2]), VECTOR([-2, 2, 1, 0, 0, 0, 2, -4]), VECTOR([-2, 2, 0, 2, 5, 2, 10, 4] ), VECTOR([-1, 2, 2, 2, 5, 2, 9, 0]), VECTOR([1, 0, 2, 1, 4, -1, 6, -7 ]), VECTOR([11, 0, 1, -14, -6, -2, -9, -8]) \}" "<+-%'VECTORG6#7*\"\"! F'\"\"\"F(F(F(F(F(-F$6#7*F(F'F(F'\"\"$,$\"\"#!\"\"\"\"&,$\"\")F/-F$6#7 *F.F,F',$\"\"%F/F,F,F2\"\"'-F$6#7*,$F(F/F.F'F'F0F'\"#6,$F.F/-F$6#7*,$F .F/F.F(F'F'F'F.,$F7F/-F$6#7*,$F.F/F.F'F.F0F.\"#5F7-F$6#7*,$F(F/F.F.F.F 0F.\"\"*F'-F$6#7*F(F'F.F(F7,$F(F/F8,$\"\"(F/-F$6#7*F=F'F(,$\"#9F/,$F8F /,$F.F/,$FMF/,$F2F/" }{TEXT 264 2 " ." }{TEXT -1 2 "\n " }{TEXT 259 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercise 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 19 1 "E(x)=exp(-x^2)" "/-%\"EG6#%\"xG-% $expG6#,$*$F&\"\"#!\"\"" }{TEXT -1 42 " . The degree n Hermite p olynomial " }{XPPEDIT 19 1 "H[n](x)" "-&%\"HG6#%\"nG6#%\"xG" }{TEXT -1 21 " is defined by " }{XPPEDIT 19 1 "H[n](x)=E(x)^(-1)*(D@@n) (E)(x)" "/-&%\"HG6#%\"nG6#%\"xG*&)-%\"EG6#F),$\"\"\"!\"\"F0---%#@@G6$% \"DGF'6#F-6#F)F0" }{TEXT -1 23 ". For example,\n\n " }{XPPEDIT 19 1 "exp(x^2)*Diff(exp(-x^2),x,x,x,x,x) = exp(x^2)*(-120*x*exp(-x^2)+ 160*x^3*exp(-x^2)-32*x^5*exp(-x^2))" "/*&-%$expG6#*$%\"xG\"\"#\"\"\"-% %DiffG6(-F%6#,$*$F(F)!\"\"F(F(F(F(F(F**&-F%6#*$F(F)F*,(*(\"$?\"F*F(F*- F%6#,$*$F(F)F2F*F2*(\"$g\"F**$F(\"\"$F*-F%6#,$*$F(F)F2F*F**(\"#KF**$F( \"\"&F*-F%6#,$*$F(F)F2F*F2F*" }{TEXT -1 13 "\n \nand so " } {XPPEDIT 19 1 "H[5](x)= -32*x^5+160*x^3-120*x" "/-&%\"HG6#\"\"&6#%\"xG ,(*&\"#K\"\"\"*$F)F'F-!\"\"*&\"$g\"F-*$F)\"\"$F-F-*&\"$?\"F-F)F-F/" } {TEXT -1 137 ". Calculate the Hermite polynomials of degree 0, 1, 2 , 3, and 4. For each n = 0, 1, 2, 3, 4, find scalars \nsuch that \+ \n\n " }{XPPEDIT 19 1 "x^n=a[0]*H[0](x)+a[1]*H[1](x)+a[2]*H[ 2](x)+a[3]*H[3](x) + a[4]*H[4](x)" "/)%\"xG%\"nG,,*&&%\"aG6#\"\"!\"\" \"-&%\"HG6#F+6#F$F,F,*&&F)6#F,F,-&F/6#F,6#F$F,F,*&&F)6#\"\"#F,-&F/6#F< 6#F$F,F,*&&F)6#\"\"$F,-&F/6#FD6#F$F,F,*&&F)6#\"\"%F,-&F/6#FL6#F$F,F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Explain why " }{XPPEDIT 265 1 "\{H[0](x), H[1](x), H[2](x), H[3](x), H[4](x) \}" "<'-&%\"HG6#\"\"!6#%\"xG-&F%6#\"\"\"6#F)-&F%6#\"\"#6#F)-&F%6#\"\" $6#F)-&F%6#\"\"%6#F)" }{TEXT 266 1 " " }{TEXT -1 91 " is a basis for t he vector space of polynomials of degree 4 or less with real coefficie nts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Informati on" }}{EXCHG {PARA 261 "" 0 "" {TEXT -1 50 "M309S04hw3R4.mws A Map leV Release 4 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 40 "Author: Brian E. Blank (02 March 2004)" }} {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This do cument may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, without" }} {PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 49 "For m ore information, please contact the author:" }}{PARA 266 "" 0 "" {TEXT -1 4 " " }}{PARA 266 "" 0 "" {TEXT -1 32 " Department of \+ Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Universi ty in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 6 3130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 267 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 56 "Copyright: \251 2004 B rian E. Blank, All Rights Reserved." }}}}}{MARK "6 6 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }