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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 11 "Math 2250-3" }}{PARA 257 "" 0 "
" {TEXT -1 18 "September 22, 2004" }}{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 90 "Coefficient matrix taken from problem \+
#19, section 3.3, page 170.  (You are assigned #20)." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 82 "A:= matrix(3,5,[2,7,-10,-19,13,1,3,-4,-8,
6,1,0,2,1,3]);\n   #the coefficient matrix" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7'\"\"#\"\"(!#5!#>\"#87'\"\"\"\"\"$
!\"%!\")\"\"'7'F0\"\"!F*F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "rref(A);  #compute the reduced row echelon form" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7%7'\"\"\"\"\"!\"\"#F(\"\"$7'F)F(!\"#!
\"$F(7'F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 310 "I want to consider
 three different linear systems for which A is the coefficient matrix.
  In the first one, the right hand sides are all zero, and  I have car
efully picked the other two right hand sides (by working backwards fro
m rref(A), actually).  The three right hand sides are the columns of t
he matrix B:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "B:=matrix(3,
3,[0,7,7,0,0,3,0,0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'ma
trixG6#7%7%\"\"!\"\"(F+7%F*F*\"\"$7%F*F*F*" }}}{PARA 0 "" 0 "" {TEXT 
-1 204 "We can consider all three linear systems at once by augmenting
 B to A, and then rref-ing: (you'll have to put in the vertical line s
eparating the coefficient matrix from the right hand sides by yourself
.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "C:=augment(A,B);\nrref
C:=rref(C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7%7*
\"\"#\"\"(!#5!#>\"#8\"\"!F+F+7*\"\"\"\"\"$!\"%!\")\"\"'F/F/F27*F1F/F*F
1F2F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rrefCG-%'matrixG6#7%7*
\"\"\"\"\"!\"\"#F*\"\"$F+F+F+7*F+F*!\"#!\"$F*F+F+F*7*F+F+F+F+F+F+F*F+
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 248 "From rref(C) you can read off the soluti
ons to each of the three problems.  Note that each column (from rref(A
)) without a leading 1 in it gives you a free parameter in the solutio
n.  Write down the solutions to our three problems on the next page:" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(C);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7%7*\"\"\"\"\"!\"\"#F(\"\"$F)F)F)7*F)F
(!\"#!\"$F(F)F)F(7*F)F)F)F)F)F)F(F)" }}}{PARA 11 "" 1 "" {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 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 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 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 31 "Important conc
eptual questions:" }}{PARA 0 "" 0 "" {TEXT -1 137 "(1) Which of these \+
three solutions could you have written down just from rrefA), rather t
han from the rref of the augmented matrix?  Why?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 376 "(2)  Linear systems in which r
ight hand side vector equals  zero are called homogeneous linear syste
ms.  Otherwise they are called inhomogeneous.  Notice that the general
 solution to the consistent inhomogeneous system is the sum of a parti
cular solution to it, together with the general solution to the homoge
neous system!!!    Was this an accident?  We'll come back to this." }}
{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 130 "By the way, Maple will go ahead
 and solve a linear system directly if you ask it to.  Here's the cons
istent inhomogeneous problem:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "b:=vector([7,3,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'
vectorG6#7%\"\"(\"\"$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "linsolve(A,b);                  " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'vectorG6#7',(*&\"\"#\"\"\"&%#_tG6#F*F*!\"\"&F,6#F)F.*&\"\"$F*&F,6
#F2F*F.,**&F)F*F+F*F**&F2F*F/F*F*F3F.F*F*F+F/F3" }}}{PARA 0 "" 0 "" 
{TEXT -1 26 "Is this the answer we got?" }}{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 0 "" }}{PARA 0 "" 0 "" {TEXT 257 68 "Using rrref(A)
 to discern general facts about the solutions to Ax=b:" }}{PARA 0 "" 
0 "" {TEXT -1 105 "The reduced row echelon form of a matrix tells you \+
a lot about possible solutions to the matrix equation " }}{PARA 0 "" 
0 "" {TEXT -1 1 "A" }{TEXT 258 1 "x" }{TEXT -1 1 "=" }{TEXT 259 1 "b" 
}{TEXT -1 48 ".  What can you say in the following situations?" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "AA:=matrix(2,5,[2, 7, -10, -
19, 13,1, 3, -4, -8, 6]);\nrref(AA);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#AAG-%'matrixG6#7$7'\"\"#\"\"(!#5!#>\"#87'\"\"\"\"\"$!\"%!\")\"\"'
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7'\"\"\"\"\"!\"\"#F(
\"\"$7'F)F(!\"#!\"$F(" }}}{PARA 259 "" 1 "" {TEXT -1 48 "Is the homoge
neous problem Ax=0 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 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 50 
"When it is solvable, how many solutions are there?" }}{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-%'matr
ixG6#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$F+F+" }}}{PARA 0 "" 0 "" 
{TEXT -1 51 "How many solutions to the homogeneous problem Bx=0?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Is the in
homogeneous problem Bx=b always solvable?   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 
"When it is solvable, how many solutions does it have?" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "C:=matrix(4,
4,[1,0,-1,1,22,-1,3,5,7,4,6,2,3,5,7,13]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"CG-%'matrixG6#7&7&\"\"\"\"\"!!\"\"F*7&\"#AF,\"\"$\"
\"&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 matrices with 1's down the dia
gonal (which runs from the upper left to lower right corner) are speci
al.  They are called identity matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 51 "How many solutions to the homogeneou
s 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 conclusions?" }}{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 ma
ny 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 -1 154 "(2)  What conditions on the di
mensions of A always force infinitely many solutions to the homogeneou
s 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 conditions 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 matrix" }}{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 "42 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }