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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 33 "One-to-One Linear Transformatio
ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 32 "Lin
ear Algebra Project          " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "
" }{TEXT 256 12 " Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 29 "Use 'lin
alg' package of Maple" }}{PARA 0 "" 0 "" {TEXT -1 58 "1.  To find whet
her a linear transformation is one-to-one." }}{PARA 0 "" 0 "" {TEXT 
-1 57 "2.  To find the inverse transformation whenever possible." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 257 21 "Required Informat
ion:" }}{PARA 0 "" 0 "" {TEXT -1 36 "If A is an  n x n square matrix a
nd " }{XPPEDIT 18 0 "T[A];" "6#&%\"TG6#%\"AG" }{TEXT -1 2 ": " }
{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 6 "   -> " }{XPPEDIT 
18 0 "R^n" "6#)%\"RG%\"nG" }{TEXT -1 25 " is multiplication by A  " }}
{PARA 0 "" 0 "" {TEXT -1 8 "   Then " }}{PARA 0 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 31 " is one-to-one
 if the range of " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 
4 " is " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }}{PARA 0 "" 0 "" 
{TEXT -1 35 "This also means A is invertible.   " }}{PARA 0 "" 0 "" 
{TEXT -1 4 "If  " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 
61 " is one-to-one then the inverse transformation is defined by " }
{XPPEDIT 18 0 "T[A^(-1)];" "6#&%\"TG6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 
10 "          " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
17 "Solved Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 8 "Let A = " }
{XPPEDIT 18 0 "matrix([[1, -2, 2], [2, 1, 1], [1, 1, 0]]);" "6#-%'matr
ixG6#7%7%\"\"\",$\"\"#!\"\"F*7%F*F(F(7%F(F(\"\"!" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "The transformation " }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 14 "
  is given by " }{XPPEDIT 18 0 "w[1];" "6#&%\"wG6#\"\"\"" }{TEXT -1 4 
" =  " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 4 " - 2" }
{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 4 " + 2" }{XPPEDIT 
18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 27 "                           " }{XPPEDIT 18 0 "w[2];" "6#&%\"wG6#
\"\"#" }{TEXT -1 5 " =  2" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 3 
" + " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 27 "                           " }{XPPEDIT 18 0 "w[
3];" "6#&%\"wG6#\"\"$" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "x[1];" "6#&%
\"xG6#\"\"\"" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"
#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "a) Determine whether
 " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 15 " is one-to-on
e." }}{PARA 0 "" 0 "" {TEXT -1 7 " b) If " }{XPPEDIT 18 0 "T[A];" "6#&
%\"TG6#%\"AG" }{TEXT -1 21 "  is one-to-one find " }{XPPEDIT 18 0 "T[A
^(-1)];" "6#&%\"TG6#)%\"AG,$\"\"\"!\"\"" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 0 "" }{TEXT 259 1 " " }{TEXT 260 9 "Solution:" }}{PARA 0 "" 
0 "" {TEXT -1 63 "a) First we check if A is invertible using the inver
se command." }}{PARA 0 "" 0 "" {TEXT -1 6 "  A = " }{XPPEDIT 18 0 "mat
rix([[1, -2, 2], [2, 1, 1], [1, 1, 0]]);" "6#-%'matrixG6#7%7%\"\"\",$
\"\"#!\"\"F*7%F*F(F(7%F(F(\"\"!" }{TEXT -1 2 "  " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "
Warning, the protected names norm and trace have been redefined and un
protected\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 49 "A := matrix([[1, -2, 2], [2, 1, 1], [1, 1, 0]]); \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"\"!\"#\"
\"#7%F,F*F*7%F*F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A
inverse:=inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)AinverseG-%
'matrixG6#7%7%\"\"\"!\"#\"\"%7%!\"\"\"\"#!\"$7%F.\"\"$!\"&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 89 "b)  The existence of the inverse of A sho
ws that T is one-to-one,  and the range of T is " }{XPPEDIT 18 0 "R^3;
" "6#*$%\"RG\"\"$" }{TEXT -1 26 ".  In fact for any vector " }}{PARA 
0 "" 0 "" {TEXT -1 5 " w = " }{XPPEDIT 18 0 "matrix([[w1], [w2], [w3]]
);" "6#-%'matrixG6#7%7#%#w1G7#%#w2G7#%#w3G" }{TEXT -1 5 "  in " }
{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 23 " there is a pre-ima
ge  " }{XPPEDIT 18 0 "matrix([[x1], [x2], [x3]]);" "6#-%'matrixG6#7%7#
%#x1G7#%#x2G7#%#x3G" }{TEXT -1 10 " given by " }{XPPEDIT 18 0 "A^(-1);
" "6#)%\"AG,$\"\"\"!\"\"" }{TEXT -1 2 "w." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "w := matrix(3,1,[w1,
 w2, w3]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'matrixG6#7%7#%
#w1G7#%#w2G7#%#w3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x= ev
alm(Ainverse&*(w));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/%\"xG-%'matrixG6#7%7#,(%#w1G\"\"\"*&\"\"#F,%#w2GF,!\"\"*&\"\"%F,%#w3G
F,F,7#,(F+F0*&F.F,F/F,F,*&\"\"$F,F3F,F07#,(F+F0*&F8F,F/F,F,*&\"\"&F,F3
F,F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The vector " }{TEXT 264 
1 "x" }{TEXT -1 22 " is given by          " }{XPPEDIT 18 0 "matrix([[x
1], [x2], [x3]]);" "6#-%'matrixG6#7%7#%#x1G7#%#x2G7#%#x3G" }{TEXT -1 
4 "  = " }{XPPEDIT 18 0 "x = matrix([[w1-2*w2+4*w3], [-w1+2*w2-3*w3], \+
[-w1+3*w2-5*w3]]);" "6#/%\"xG-%'matrixG6#7%7#,(%#w1G\"\"\"*&\"\"#F,%#w
2GF,!\"\"*&\"\"%F,%#w3GF,F,7#,(F+F0*&F.F,F/F,F,*&\"\"$F,F3F,F07#,(F+F0
*&F8F,F/F,F,*&\"\"&F,F3F,F0" }{TEXT -1 4 "    " }}}}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 261 18 "
 Solved Example 2:" }}{PARA 0 "" 0 "" {TEXT -1 9 " Let A = " }
{XPPEDIT 18 0 "matrix([[1, 2, 1], [-2, 1, 4], [7, 4, -5]]);" "6#-%'mat
rixG6#7%7%\"\"\"\"\"#F(7%,$F)!\"\"F(\"\"%7%\"\"(F-,$\"\"&F," }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "
  " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 14 "  is given b
y " }{XPPEDIT 18 0 "w[1];" "6#&%\"wG6#\"\"\"" }{TEXT -1 4 " =  " }
{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 4 " + 4" }{XPPEDIT 
18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "x[3];
" "6#&%\"xG6#\"\"$" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "    \+
                      " }{XPPEDIT 18 0 "w[2];" "6#&%\"wG6#\"\"#" }
{TEXT -1 5 " =  2" }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT 
-1 3 " +7" }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 3 " + " 
}{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 27 "                           " }{XPPEDIT 18 0 "w[3];" 
"6#&%\"wG6#\"\"$" }{TEXT -1 4 " =  " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6
#\"\"\"" }{TEXT -1 3 " +3" }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 " a) Show that  " }
{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 19 " is not one-to-on
e." }}{PARA 0 "" 0 "" {TEXT -1 28 " b) Find  all vectors x in  " }
{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 36 " which are mapped \+
onto the 0 vector." }}{PARA 0 "" 0 "" {TEXT -1 44 "(Note that: For a o
ne-to-one transformation " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }
{TEXT -1 46 " only the 0- vector is mapped onto 0 vector.  " }}{PARA 
0 "" 0 "" {TEXT -1 26 "In other words Ax = 0 or A" }{XPPEDIT 18 0 "mat
rix([[x[1]], [x[2]], [x[3]]]);" "6#-%'matrixG6#7%7#&%\"xG6#\"\"\"7#&F)
6#\"\"#7#&F)6#\"\"$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[0], [0
], [0]]);" "6#-%'matrixG6#7%7#\"\"!7#F(7#F(" }{TEXT -1 31 " has only t
he trivial solution " }}{PARA 0 "" 0 "" {TEXT -1 82 "or the solution s
pace of the system of corresponding linear equations consists of " }
{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 5 "= 0, " }
{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 5 "= 0, " }{XPPEDIT 
18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 3 "= 0" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 263 10 " Solution:" }}
{PARA 0 "" 0 "" {TEXT -1 32 "(a) We check whether matrix A = " }
{XPPEDIT 18 0 "matrix([[1, 2, 1], [-2, 1, 4], [7, 4, -5]]);" "6#-%'mat
rixG6#7%7%\"\"\"\"\"#F(7%,$F)!\"\"F(\"\"%7%\"\"(F-,$\"\"&F," }{TEXT 
-1 15 " is invertible." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:
= matrix(3,3, [1,2,1,-2,1,4,7,4,-5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'matrixG6#7%7%\"\"\"\"\"#F*7%!\"#F*\"\"%7%\"\"(F.!\"&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}{PARA 8 "" 1 
"" {TEXT -1 36 "Error, (in inverse) singular matrix\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "This means  matrix  A is not invertible.
" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 32 "(b)
 Now we find all vector w in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }
{TEXT -1 31 " which map onto the 0 vector.  " }}{PARA 0 "" 0 "" {TEXT 
-1 81 "We do this by finding  the rref of the system Ax = 0 using the \+
augmented matrix. " }}{PARA 0 "" 0 "" {TEXT -1 41 "                   \+
               Ax = 0 " }}{PARA 0 "" 0 "" {TEXT -1 6 "or    " }
{XPPEDIT 18 0 "matrix([[1, 2, 1], [-2, 1, 4], [7, 4, -5]]);" "6#-%'mat
rixG6#7%7%\"\"\"\"\"#F(7%,$F)!\"\"F(\"\"%7%\"\"(F-,$\"\"&F," }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "matrix([[x1], [x2], [x3]]);" "6#-%'matrixG6#7
%7#%#x1G7#%#x2G7#%#x3G" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[0],
 [0], [0]]);" "6#-%'matrixG6#7%7#\"\"!7#F(7#F(" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "w := matrix(3,1,[0,0,0]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"wG-%'matrixG6#7%7#\"\"!F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "Aw := augment(A,w);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#AwG-%'matrixG6#7%7&\"\"\"\"\"#F*\"\"!7&!\"#F*\"\"%F,
7&\"\"(F/!\"&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Awreduce
d:= rref(Aw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*AwreducedG-%'matri
xG6#7%7&\"\"\"\"\"!#!\"(\"\"&F+7&F+F*#\"\"'F.F+7&F+F+F+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "This means that the solution space consis
ts of vectors " }{XPPEDIT 18 0 "matrix([[x[1]], [x[2]], [x[3]]]);" "6#
-%'matrixG6#7%7#&%\"xG6#\"\"\"7#&F)6#\"\"#7#&F)6#\"\"$" }{TEXT -1 7 " \+
where " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "7*x[3]/5;" "6#*(\"\"(\"\"\"&%\"xG6#\"\"$F%\"\"&!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 3 "
 = " }{XPPEDIT 18 0 "6*x[3]/5;" "6#*(\"\"'\"\"\"&%\"xG6#\"\"$F%\"\"&!
\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "or all vectors   \+
" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{XPPEDIT 18 0 "matrix([[7/5]
, [6/5], [1]]);" "6#-%'matrixG6#7%7#*&\"\"(\"\"\"\"\"&!\"\"7#*&\"\"'F*
F+F,7#F*" }{TEXT -1 8 "  where " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG
" }{TEXT -1 13 " is a scalar." }}}}{PARA 0 "" 0 "" {TEXT -1 74 "______
____________________________________________________________________" 
}}{PARA 257 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT 
-1 41 "In each of the problems for the matrix A " }}{PARA 0 "" 0 "" 
{TEXT -1 11 " a) define " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 " b) determine whether " 
}{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 15 " is one-to-one.
" }}{PARA 0 "" 0 "" {TEXT -1 7 " c) if " }{XPPEDIT 18 0 "T[A];" "6#&%
\"TG6#%\"AG" }{TEXT -1 54 "  is one-to-one find the inverse transforma
tion.  If  " }{XPPEDIT 18 0 "T[A]" "6#&%\"TG6#%\"AG" }{TEXT -1 71 " is
 not one-to-one find all the vectors which map onto the 0 vector.   " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 "
 " }{TEXT 265 10 "Problem 1:" }}{PARA 0 "" 0 "" {TEXT -1 5 " A = " }
{XPPEDIT 18 0 "matrix([[1, 3, -4], [2, 7, 1], [1, 3, 0]]);" "6#-%'matr
ixG6#7%7%\"\"\"\"\"$,$\"\"%!\"\"7%\"\"#\"\"(F(7%F(F)\"\"!" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 1 " " }{TEXT 266 10 "Problem 2:" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "A = " }{XPPEDIT 18 0 "matrix([[2, 0, -4], [2, 5, -1], [-1,
 3, 0]]);" "6#-%'matrixG6#7%7%\"\"#\"\"!,$\"\"%!\"\"7%F(\"\"&,$\"\"\"F
,7%,$F0F,\"\"$F)" }}}{PARA 0 "" 0 "" {TEXT -1 77 "____________________
_________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 172 "MSIP Grant #P120A80089-98:  \"Three Urban Calculu
s Reform programs: Adopting the Best\" 1998-2001, MSEIP Grant #P120A01
0031: \"Four Colleges: Calculus + Enhancements\"  2001-04" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{MARK "22 0" 97 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }