{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 19 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 287 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 289 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 19 291 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 293 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 19 295 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 303 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 314 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Norm al" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 " Math 309 Fall 2001 Exam \+ 1" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "1. " }{TEXT 256 5 " Is " } {XPPEDIT 257 1 "\{[x,y,2*x]*` : where `*x+3*y=0" "<#/,&*(7%%\"xG%\"yG *&\"\"#\"\"\"F'F+F+%+~:~~where~GF+F'F+F+*&\"\"$F+F(F+F+\"\"!" }{TEXT 258 43 " a vector space? Explain why or why not." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 17 "Solution: It is!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "To justi fy our positive answer we must verify that the given set is closed und er addition and scalar multiplication:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Closed under addition: " }{XPPEDIT 18 0 "VECTOR([x[1], y[1], 2*x[1]])+VECTOR([x[2], y[2], 2*x[2]]) = VECT OR([x[1]+x[2], y[1]+y[2], 2*(x[1]+x[2])])" "/,&-%'VECTORG6#7%&%\"xG6# \"\"\"&%\"yG6#F+*&\"\"#F+&F)6#F+F+F+-F%6#7%&F)6#F0&F-6#F0*&F0F+&F)6#F0 F+F+-F%6#7%,&&F)6#F+F+&F)6#F0F+,&&F-6#F+F+&F-6#F0F+*&F0F+,&&F)6#F+F+&F )6#F0F+F+" }}{PARA 0 "" 0 "" {TEXT -1 6 "and " }{XPPEDIT 19 1 "x[1]+ x[2], y[1]+3*(y[1]+y[2]) = [x[1]+3y[1]]+[x[2]+3*y[2]]*`=0+0=0`" "6$,&& %\"xG6#\"\"\"F'&F%6#\"\"#F'/,&&%\"yG6#F'F'*&\"\"$F',&&F.6#F'F'&F.6#F*F 'F'F',&7#,&&F%6#F'F'*&F1F'&F.6#F'F'F'F'*&7#,&&F%6#F*F'*&F1F'&F.6#F*F'F 'F'%'=0+0=0GF'F'" }}{PARA 0 "" 0 "" {TEXT -1 65 "Therefore, the sum of two elements in the set is also in the set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Closed under multiplication by \+ scalars: " }{XPPEDIT 19 1 "c*[x,y,2*x]=[c*x,c*y,c*2*x]" "/*&%\"cG\" \"\"7%%\"xG%\"yG*&\"\"#F%F'F%F%7%*&F$F%F'F%*&F$F%F(F%*(F$F%F*F%F'F%" } }{PARA 0 "" 0 "" {TEXT -1 11 "and since " }{XPPEDIT 19 1 "c*x+3*c*y=c *(x+3*y)" "/,&*&%\"cG\"\"\"%\"xGF&F&*(\"\"$F&F%F&%\"yGF&F&*&F%F&,&F'F& *&F)F&F*F&F&F&" }{TEXT -1 13 " is 0 when " }{XPPEDIT 19 1 "x+3*y=0" "/,&%\"xG\"\"\"*&\"\"$F%%\"yGF%F%\"\"!" }{TEXT -1 4 " , " }{XPPEDIT 19 1 "c*[x,y,2*x]" "*&%\"cG\"\"\"7%%\"xG%\"yG*&\"\"#F$F&F$F$" }{TEXT -1 30 " is in the given set when " }{XPPEDIT 19 1 "[x,y,2*x]" "7%% \"xG%\"yG*&\"\"#\"\"\"F#F'" }{TEXT -1 5 " is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 2 "2." }{TEXT 265 56 " Find and suitably describe all solu tions of the system " }{XPPEDIT 266 1 "MATRIX([[1, 0, 1, 1, 2], [1, 1, 0, 1, 1], [1, 2, 1, 0, 1]])*MATRIX([[x], [y], [z], [u], [v]])=MATRIX( [[0], [0], [0])" "/*&-%'MATRIXG6#7%7'\"\"\"\"\"!F)F)\"\"#7'F)F)F*F)F)7 'F)F+F)F*F)F)-F%6#7'7#%\"xG7#%\"yG7#%\"zG7#%\"uG7#%\"vGF)-F%6#7%7#F*7# F*7#F*" }{TEXT 267 1 "." }}{PARA 0 "" 0 "" {TEXT 300 9 "Solution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "One way t o do this is by row reduction:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A := matrix([[1, 0, 1, 1, 2] , [1, 1, 0, 1, 1], [1, 2, 1, 0, 1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7'\"\"\"\"\"!F*F*\"\"#7'F*F*F+F*F*7'F*F,F*F+F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7'\"\"\"\"\"!F)#\"\"$\"\"#F*7'F)F(F) #!\"\"F,F.7'F)F)F(F.#F(F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "This tells us that we can choose " }{XPPEDIT 19 1 "u" "I\"uG6\"" }{TEXT -1 7 " and " }{XPPEDIT 19 1 "v" "I\"vG6\"" } {TEXT -1 25 " arbitrarily after which" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "x=-3/2*u-3/2 *v" "/%\"xG,&*(\"\"$\"\"\"\"\"#!\"\"%\"uGF'F)*(F&F'F(F)%\"vGF'F)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "y=1/2*u+1/2*v" "/%\"yG,&*(\"\"\"F&\"\"#!\"\"%\"uGF&F&* (F&F&F'F(%\"vGF&F&" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 19 1 "z = 1/2*u-1/2*v" "/%\"zG,&*( \"\"\"F&\"\"#!\"\"%\"uGF&F&*(F&F&F'F(%\"vGF&F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "Another way to write the same thing, using vectors, is" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 19 1 "MATRIX([[x],[y],[z], [u],[v]])=u*MATRIX([[-3/2],[1/2],[1/2],[1],[0]])+v*MATRIX([[-3/2],[1/2 ],[-1/2],[0],[1]])" "/-%'MATRIXG6#7'7#%\"xG7#%\"yG7#%\"zG7#%\"uG7#%\"v G,&*&F.\"\"\"-F$6#7'7#,$*&\"\"$F3\"\"#!\"\"F<7#*&F3F3F;F<7#*&F3F3F;F<7 #F37#\"\"!F3F3*&F0F3-F$6#7'7#,$*&F:F3F;F*MATRIX([[1/2],[sqrt(3)/2]]))" "/-%\"PG6#-%'M ATRIXG6#7$7#%\"xG7#%\"yG*&-%-anglebracketG6$-F'6#7$7#F+7#F--F'6#7$7#*& \"\"\"F<\"\"#!\"\"7#*&-%%sqrtG6#\"\"$FF<-F'6#7$7#*&F7#*&-F B6#FDFF<" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "where the angle brackets denote the \"dot\" or scalar pro duct." }}{PARA 0 "" 0 "" {TEXT -1 23 "Expanding this gives us" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 19 1 "P(MATRIX([[x],[y]]))= MATRIX( [ [x/4+sqrt(3)/4*y], [ s qrt(3)/4*x+y/4 ] ] )" "/-%\"PG6#-%'MATRIXG6#7$7#%\"xG7#%\"yG-F'6#7$7#, &*&F+\"\"\"\"\"%!\"\"F4*(-%%sqrtG6#\"\"$F4F5F6F-F4F47#,&*(-F96#F;F4F5F 6F+F4F4*&F-F4F5F6F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 19 1 "P(MATRIX([[x],[y]]))= MATRIX( [ [1 /4,sqrt(3)/4], [ sqrt(3)/4,1/4 ] ] )*MATRIX([[x],[y]])" "/-%\"PG6#-%'M ATRIXG6#7$7#%\"xG7#%\"yG*&-F'6#7$7$*&\"\"\"F4\"\"%!\"\"*&-%%sqrtG6#\" \"$F4F5F67$*&-F96#F;F4F5F6*&F4F4F5F6F4-F'6#7$7#F+7#F-F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "and so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 19 1 "A = MATRIX( [ [1/4,sqrt(3)/4], [ sqrt (3)/4,1/4 ] ] )" "/%\"AG-%'MATRIXG6#7$7$*&\"\"\"F*\"\"%!\"\"*&-%%sqrtG 6#\"\"$F*F+F,7$*&-F/6#F1F*F+F,*&F*F*F+F," }}{PARA 0 "" 0 "" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "4. " }{TEXT 259 28 "Calculate the inverse of " } {XPPEDIT 260 1 "MATRIX([[1, 2, 2], [1, 1, 1], [1, 0, -1]])" "-%'MATRIX G6#7%7%\"\"\"\"\"#F(7%F'F'F'7%F'\"\"!,$F'!\"\"" }{TEXT 261 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 10 "Solution : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The calculation is standard. Here is the result:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "inverse(matr ix([[1, 2, 2], [1, 1, 1], [1, 0, -1]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%!\"\"\"\"#\"\"!7%F)!\"$\"\"\"7%F(F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 3 "5. " }{TEXT 262 19 " Solve the system " } {XPPEDIT 263 1 "MATRIX([[1, 0, 0], [1, 1, 0], [1, 2, 1]])*MATRIX([[1, \+ 0, -1], [0, 2, 3], [0, 0, 1/2]])*MATRIX([[x], [y], [z]])=MATRIX([[1], \+ [2], [3]])" "/*(-%'MATRIXG6#7%7%\"\"\"\"\"!F*7%F)F)F*7%F)\"\"#F)F)-F%6 #7%7%F)F*,$F)!\"\"7%F*F-\"\"$7%F*F**&F)F)F-F3F)-F%6#7%7#%\"xG7#%\"yG7# %\"zGF)-F%6#7%7#F)7#F-7#F5" }{TEXT 264 64 " by using the operations o f forward and backward substitution. " }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 306 9 "Solut ion:" }}{PARA 0 "" 0 "" {TEXT -1 48 "By forward substitution we see th e solution of " }{XPPEDIT 19 1 "MATRIX([[1, 0, 0], [1, 1, 0], [1, 2, \+ 1]])*MATRIX([[x], [y], [z]])=MATRIX([[1], [2], [3]])" "/*&-%'MATRIXG6# 7%7%\"\"\"\"\"!F*7%F)F)F*7%F)\"\"#F)F)-F%6#7%7#%\"xG7#%\"yG7#%\"zGF)-F %6#7%7#F)7#F-7#\"\"$" }{TEXT -1 4 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 19 1 "x=1" "/%\"xG\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 19 1 "y=1" "/%\"yG\"\"\"" } }{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 19 1 "z=0" "/%\"zG\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "and by backward substitution the solution of " }{XPPEDIT 19 1 "MATRIX([[1, 0, -1], [0, 2, 3], [0, 0, 1/2]])*M ATRIX([[x], [y], [z]])=MATRIX([[1], [1], [0]])" "/*&-%'MATRIXG6#7%7%\" \"\"\"\"!,$F)!\"\"7%F*\"\"#\"\"$7%F*F**&F)F)F.F,F)-F%6#7%7#%\"xG7#%\"y G7#%\"zGF)-F%6#7%7#F)7#F)7#F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 19 1 "z=0" "/%\"zG\"\"!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 19 1 "y=1/2" "/%\"yG*&\"\"\"F%\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 19 1 "x = 1 " "/%\"xG\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Most forward and backward substitutions can be done in on e's head. Here is a verification: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "linsolve(matrix([[1, 0, 0], [1, 1, 0], [1, 2, 1]]) &* matrix([[ 1, 0, -1], [0, 2, 3], [0, 0, 1/2]]) , vector([1, 2, 3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"\"#F'\"\"#\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 " 6. " } {TEXT 268 23 "Find a linear map from " }{XPPEDIT 269 1 "R^2" "*$%\"RG \"\"#" }{TEXT 270 4 " to " }{XPPEDIT 271 1 "R^3" "*$%\"RG\"\"$" } {TEXT 272 18 " with the plane " }{XPPEDIT 273 1 "\{[x,y,z]*` : x - \+ y + 2z = 0`\}" "<#*&7%%\"xG%\"yG%\"zG\"\"\"%3~~:~x~-~y~+~2z~=~0GF(" } {TEXT 274 15 " as its image." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 310 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "The vectors " }{XPPEDIT 19 1 "MATRIX( [[1],[1],[0]])" "-%'MATRIXG6#7%7#\"\"\"7#F'7#\"\"!" }{TEXT -1 8 " an d " }{XPPEDIT 19 1 "MATRIX([[0],[2],[1]])" "-%'MATRIXG6#7%7#\"\"!7#\" \"#7#\"\"\"" }{TEXT -1 111 " are in the plane since their components satisfy its cartesian equation. Therefore every linear combination \+ " }{XPPEDIT 19 1 "s*MATRIX([[1],[1],[0]])+t*MATRIX([[0],[2],[1]])" ",& *&%\"sG\"\"\"-%'MATRIXG6#7%7#F%7#F%7#\"\"!F%F%*&%\"tGF%-F'6#7%7#F-7#\" \"#7#F%F%F%" }{TEXT -1 64 " is also in the plane. We can write this l inear combination as " }{XPPEDIT 19 1 "MATRIX([[1,0],[1,2],[0,1]])*MAT RIX([[s],[t]])" "*&-%'MATRIXG6#7%7$\"\"\"\"\"!7$F(\"\"#7$F)F(F(-F$6#7$ 7#%\"sG7#%\"tGF(" }{TEXT -1 25 ". Therefore we can take " }{XPPEDIT 19 1 "T(MATRIX([[s],[t]]))=MATRIX([[1,0],[1,2],[0,1]])*MATRIX([[s],[t] ])" "/-%\"TG6#-%'MATRIXG6#7$7#%\"sG7#%\"tG*&-F'6#7%7$\"\"\"\"\"!7$F3\" \"#7$F4F3F3-F'6#7$7#F+7#F-F3" }{TEXT -1 28 " as the required linear ma p." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 " 7. " }{TEXT 275 23 "Find a line ar map from " }{XPPEDIT 276 1 "R^3" "*$%\"RG\"\"$" }{TEXT 277 4 " to \+ " }{XPPEDIT 278 1 "R^1" "*$%\"RG\"\"\"" }{TEXT 279 18 " with the plan e " }{XPPEDIT 280 1 "\{[x,y,z]*` : x - y + 2z = 0`\}" "<#*&7%%\"xG% \"yG%\"zG\"\"\"%3~~:~x~-~y~+~2z~=~0GF(" }{TEXT 281 16 " as its kernel ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 9 "Solu tion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 19 1 "T(MATRIX([[x],[y],[z]]))=x - y + 2*z " "/-%\"TG 6#-%'MATRIXG6#7%7#%\"xG7#%\"yG7#%\"zG,(F+\"\"\"F-!\"\"*&\"\"#F1F/F1F1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "does \+ the trick." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 4 " 8. " }{TEXT 296 14 "Suppose that " }{XPPEDIT 295 1 "T( MATRIX([[x],[y]]))=MATRIX([[1,2],[3,6]])*MATRIX([[x],[y]])" "/-%\"TG6# -%'MATRIXG6#7$7#%\"xG7#%\"yG*&-F'6#7$7$\"\"\"\"\"#7$\"\"$\"\"'F3-F'6#7 $7#F+7#F-F3" }{TEXT 297 55 ". Describe the image and kernel of this tr ansformation." }}{PARA 0 "" 0 "" {TEXT 312 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The kernel of " } {XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 45 " is the same as the kernel of the rref of " }{XPPEDIT 19 1 "MATRIX([[1,2],[3,6]])" "-%'MATRIXG 6#7$7$\"\"\"\"\"#7$\"\"$\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A := matrix( [[1, 2], [3, 6]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG 6#7$7$\"\"\"\"\"#7$\"\"$\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7$\"\" \"\"\"#7$\"\"!F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The kernel consists of all scalar multiples of " } {XPPEDIT 19 1 "MATRIX([[-2],[1]])" "-%'MATRIXG6#7$7#,$\"\"#!\"\"7#\"\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The image of " }{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 195 " consists of all linear combinations of the columns of the giv en matrix. But the second column is twice the first so all linear comb inations of the columns are multiples of the first column, " } {XPPEDIT 19 1 "MATRIX([[1],[3]])" "-%'MATRIXG6#7$7#\"\"\"7#\"\"$" } {TEXT -1 1 "." }}}}{MARK "3 19 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }