{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 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 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2250-3" }}{PARA 257 "" 0 " " {TEXT -1 18 "September 17, 2003" }}{PARA 258 "" 0 "" {TEXT -1 20 "Ma ple linear algebra" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "with(linalg): #load linear algebra package" }}}{PARA 0 "" 0 "" {TEXT -1 65 "Problem #17, section 3.2, page 160. T his was a homework problem." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A:= matrix(3,4,[1,-4,-3,-3,2,-6,-5,-5,3,-1,-4,-5]);\n #the coeff icient matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7 &\"\"\"!\"%!\"$F,7&\"\"#!\"'!\"&F07&\"\"$!\"\"F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b:=vector([2,5,-7]); #the right hand side " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'vectorG6#7%\"\"#\"\"&!\"( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Aaugb:=augment(A,b); # the augmented matrix...\n #you could have entered it directly too." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AaugbG-%'matrixG6#7%7'\"\"\"!\"%! \"$F,\"\"#7'F-!\"'!\"&F0\"\"&7'\"\"$!\"\"F+F0!\"(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 51 "rref(Aaugb); #compute the reduced row echel on form" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7'\"\"\"\"\"! F)\"\"#\"#T7'F)F(F)!\"\"!#=7'F)F)F(\"\"$\"#P" }}}{PARA 0 "" 0 "" {TEXT -1 277 "From rref(Aaugb) you can read off the solution to the or iginal problem. Each column (from A) without a leading 1 gives you a \+ free parameter in the solution. so we see that x4=t, x3=37-3t, x2=-1 8+t, x1=41-2t is the general solution. We could also write this in v ector form:" }}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7&7 #&%\"xG6#\"\"\"7#&F*6#\"\"#7#&F*6#\"\"$7#&F*6#\"\"%,&-F%6#7&7#\"#T7#!# =7#\"#P7#\"\"!F,*&%\"tGF,-F%6#7&7#!\"#7#F,7#!\"$FLF,F," }}}{PARA 0 "" 0 "" {TEXT 256 30 "Important conceptual question:" }}{PARA 0 "" 0 "" {TEXT -1 549 "Linear systems in which right hand side vector equals z ero are called homogeneous linear systems. Otherwise they are called \+ inhomogeneous. Notice that in the problem above, the constant vector \+ [41,-18,37,0] is a particular solution to the inhomogeneous system. \+ Notice that the set of multiples \{t*[-2,1,-3,1]\} is the general solu tion to the homogeneous problem. (Why?) So the general solution to o ur system was a particular solution to it plus the general solution to the homogeneous equation. Was this an accident? We'll come back to \+ this." }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 86 "By the way, Maple will \+ go ahead and solve the linear system directly if you ask it to:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "linsolve(A,b); \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&,&\"#T\"\"\"*&\" \"#F)&%#_tG6#F)F)!\"\",&\"#=F/F,F),&\"#PF)*&\"\"$F)F,F)F/F," }}}{PARA 0 "" 0 "" {TEXT -1 40 "Would this have been the same answer as:" }} {EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&\"\"&\"\"\"*&\"\"#F&%\"tGF& !\"\"F),&\"# " 0 "" {MPLTEXT 1 0 54 "A:=matrix(3,5,[2,7,-10,-19,13,1,3,-4,-8,6,1,0,2,1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'\"\"#\"\"(!#5!#>\"#87' \"\"\"\"\"$!\"%!\")\"\"'7'F0\"\"!F*F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "RREFA:=rref(A); #the actual command is lower case!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RREFAG-%'matrixG6#7%7'\"\"\"\"\"! \"\"#F*\"\"$7'F+F*!\"#!\"$F*7'F+F+F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 94 "How many solutions to the homogeneous problem Ax=0? How many free parameters in the solution?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Is the inhomogeneous problem Ax=b always solvable? \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "When it is solvable, how many solutions are the re?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "B:=matrix(3,2,[1,2,-1,3,4,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7$\"\"\"\"\"#7$!\"\"\"\"$7$\"\"%F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "RREFB:=rref(B);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RREFBG-%'matrixG6#7%7$\"\"\"\"\"!7$ F+F*7$F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 51 "How many solutions to the \+ homogeneous problem Bx=0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 53 "Is the inhomogeneous problem Bx=b always solvable? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 53 "When it is solvable, how many solutions \+ does it have?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "C:=matrix(4,4,[1,0,-1,1,22,-1,3,5,7,4,6,2,3,5,7,13 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7&7&\"\"\"\" \"!!\"\"F*7&\"#AF,\"\"$\"\"&7&\"\"(\"\"%\"\"'\"\"#7&F/F0F2\"#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "RREFC:=rref(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RREFCG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F +F+7&F+F+F*F+7&F+F+F+F*" }}}{PARA 0 "" 0 "" {TEXT -1 146 "Square matri ces with 1's down the diagonal (which runs from the upper left to lowe r right corner) are special. They are called identity matrices." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "How many \+ solutions to the homogeneous problem Cx=0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Is the inhomogeneous problem Cx =b always solvable? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "How many solutions?" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "What are your general conclusio ns?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "( 1) What conditions on rref(A) guarantee that the homogeneous equation Ax=0 has infinitely many solutions?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "(2) Wha t conditions on the dimensions of A force infinitely many solutions to the homogeneous problem regardless of the individual entries of A?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "(3) What cond itions on rref(A) guarantee that solutions x to Ax=b are always unique (if they exist)?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "(4) If A is a square matrix, what can you say about solutions to Ax= b when" }}{PARA 0 "" 0 "" {TEXT -1 36 "(4a) rref(A) is the identity m atrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "(4b) \+ rref(A) is not the identity matrix?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "77 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }