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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 31 "Coordinates and Change \+
of Basis" }}{PARA 258 "" 0 "" {TEXT 256 4 "HW 5" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+]
  sign to expand a section. Click on a [-] sign to collapse a section.
 To do these exercises you 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 the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 20 "Stud
ent Name and ID:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this  \+
" }{TEXT 257 5 "MAPLE" }{TEXT -1 3 "   " }{HYPERLNK 17 "worksheet" 2 "
worksheet" "" }{TEXT -1 108 ",  you will be asked to  to perform calcu
lations pertaining to change of basis and coordinates. You may use " }
{TEXT 266 5 "MAPLE" }{TEXT -1 57 "  to perform the calculations or you
 may do them my hand." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports
" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you prepare with " }
{TEXT 267 5 "MAPLE" }{TEXT -1 112 "   should be prepared with the same
 care that you would devote to laboratory reports in biology and chemi
stry.  " }{TEXT 269 46 "A report should not be a diary or history of a
" }{TEXT -1 1 " " }{TEXT 268 6 " MAPLE" }{TEXT -1 2 "  " }{TEXT 270 1 
" " }{TEXT 271 51 "session.  Delete what is not needed for the report.
" }{TEXT -1 25 "  All lines of the form  " }{TEXT 259 6 "?topic" }
{TEXT -1 82 "  (that arise from help queries) should be erased.  All e
rrors should be erased.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 272 5 "MAPLE" }
{TEXT -1 165 "  report, think about the toner and paper resources that
 you are using. All commands must be terminated - either with the stan
dard terminator, the semicolon, or the " }{TEXT 276 17 "silent termina
tor" }{TEXT -1 57 ", the colon. When you assign a variable, for exampl
e  \n \"" }{TEXT 277 7 "x := 5;" }{TEXT -1 30 " \",  there is no need \+
to have " }{TEXT 273 5 "MAPLE" }{TEXT -1 12 " echo back  " }{TEXT 260 
6 "x := 5" }{TEXT -1 89 ".  When this is printed, it simply wastes pap
er and ink.  Choose the silent terminator  \"" }{TEXT 261 7 "x := 5:" 
}{TEXT -1 70 " \"  instead.  When you load a package (without the sile
nt terminator)," }{TEXT 262 1 " " }{TEXT -1 2 "  " }{TEXT 274 5 "MAPLE
" }{TEXT -1 123 " will list the commands that become available with th
e package. This is fine - it will help you become familiar with what  \+
" }{TEXT 275 5 "MAPLE" }{TEXT -1 125 " makes available. However, these
 commands should not be part of a lab report. Reload the package with \+
the silent terminator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 79 "Much of the text in this worksheet should be deleted
.  For example, delete the " }{TEXT 263 12 "Introduction" }{TEXT -1 5 
" and " }{TEXT 264 8 "Keywords" }{TEXT -1 34 " sections. Delete this s
ection on " }{TEXT 265 7 "Reports" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your works
heet should execute in the order that it has been written.  In particu
lar, remember that the ditto refers to the result of the last executed
 command - not the result of the command that physically precedes it. \+
 " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Keywords" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Clicking 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[a
ddrow]" "" }{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 "ditto" 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 "linalg" "" }{TEXT -1 2 ", \+
" }}{PARA 15 "" 0 "" {TEXT -1 3 "   " }{HYPERLNK 17 "linsolve" 2 "lins
olve" "" }{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 "scalarmu
l" 2 "scalarmul" "" }{TEXT -1 1 "," }}{PARA 15 "" 0 "" {TEXT -1 3 "   \+
" }{HYPERLNK 17 "swaprow" 2 "linalg[swaprow]" "" }{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 "w
ith" 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 0 "" 0 "" {TEXT -1 25 "Backgroun
d worksheet:    " }{HYPERLNK 17 "transitionMatrixR4.mws" 1 "rankR4.mws
" "" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Let     " }{XPPEDIT 
19 1 "u[1]=MATRIX([[ 1], [ 0], [2], [-1]])" "/&%\"uG6#\"\"\"-%'MATRIXG
6#7&7#F&7#\"\"!7#\"\"#7#,$F&!\"\"" }{TEXT -1 6 ",     " }{XPPEDIT 19 
1 "u[2]=MATRIX([[ 1], [ 0], [0], [-1]])" "/&%\"uG6#\"\"#-%'MATRIXG6#7&
7#\"\"\"7#\"\"!7#F.7#,$F,!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 19 1 "u[
3]=MATRIX([[ 0], [ 1], [0], [2]])" "/&%\"uG6#\"\"$-%'MATRIXG6#7&7#\"\"
!7#\"\"\"7#F,7#\"\"#" }{TEXT -1 12 ",   and     " }{XPPEDIT 19 1 "u[4]
=MATRIX([[ 3], [ 1], [-1], [0]])" "/&%\"uG6#\"\"%-%'MATRIXG6#7&7#\"\"$
7#\"\"\"7#,$F.!\"\"7#\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "
Let     " }{XPPEDIT 19 1 "v[1]=MATRIX([[ 1], [ 1], [0], [0]])" "/&%\"v
G6#\"\"\"-%'MATRIXG6#7&7#F&7#F&7#\"\"!7#F." }{TEXT -1 6 ",     " }
{XPPEDIT 19 1 "v[2]=MATRIX([[ 1], [ 0], [0], [1]])" "/&%\"vG6#\"\"#-%'
MATRIXG6#7&7#\"\"\"7#\"\"!7#F.7#F," }{TEXT -1 5 ",    " }{XPPEDIT 19 
1 "v[3]=MATRIX([[ 0], [ 1], [0], [-2]])" "/&%\"vG6#\"\"$-%'MATRIXG6#7&
7#\"\"!7#\"\"\"7#F,7#,$\"\"#!\"\"" }{TEXT -1 12 ",   and     " }
{XPPEDIT 19 1 "u[4]=MATRIX([[ 1], [ 1], [-1], [1]])" "/&%\"uG6#\"\"%-%
'MATRIXG6#7&7#\"\"\"7#F,7#,$F,!\"\"7#F," }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "The coordinates of a vector   " }{XPPEDIT 19 1 "w" "I\"wG
6\"" }{TEXT -1 34 "   with respect to the basis      " }{XPPEDIT 19 1 
"[u[1],u[2],u[3],u[4]]" "7&&%\"uG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"
%" }{TEXT -1 13 "    are      " }{XPPEDIT 19 1 "MATRIX([[ 2], [ 3], [-
1], [1]])" "-%'MATRIXG6#7&7#\"\"#7#\"\"$7#,$\"\"\"!\"\"7#F," }{TEXT 
-1 5 " . \n\n" }}{PARA 0 "" 0 "" {TEXT 280 5 "( i )" }{TEXT -1 61 "   \+
 Compute the transition matrix from  the ordered basis    " }{XPPEDIT 
19 1 "[u[1],u[2],u[3],u[4]]" "7&&%\"uG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6
#\"\"%" }{TEXT -1 30 "    to the ordered basis      " }{XPPEDIT 19 1 "
[v[1],v[2],v[3],v[4]]" "7&&%\"vG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%
" }{TEXT -1 4 " . \n" }{TEXT 281 7 "( ii  )" }{TEXT -1 55 "  Use part \+
( i )  to calculate  the coordinates of     " }{XPPEDIT 19 1 "w" "I\"w
G6\"" }{TEXT -1 34 "   with respect to the basis      " }{XPPEDIT 19 
1 "[v[1],v[2],v[3],v[4]]" "7&&%\"vG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"
\"%" }{TEXT -1 4 " .  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 13 "Exercise \+
2   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "B
ackground worksheet:      " }{HYPERLNK 17 "transitionMatrixR4.mws" 1 "
transitionMatrixR4.mws" "" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let    " }{XPPEDIT 19 1 "[u[1],u[2],u[
3],u[4]]" "7&&%\"uG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 
9 "  and    " }{XPPEDIT 19 1 "[v[1],v[2],v[3],v[4]]" "7&&%\"vG6#\"\"\"
&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 23 "   be as in Exercise 1.
" }}{PARA 0 "" 0 "" {TEXT -1 26 "A linear transformation   " }
{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 9 "   from  " }{XPPEDIT 19 1 "R
^4" "*$%\"RG\"\"%" }{TEXT -1 6 "  to  " }{XPPEDIT 19 1 "R^4" "*$%\"RG
\"\"%" }{TEXT -1 21 "     has matrix      " }{XPPEDIT 19 1 "MATRIX([[0
, 0, 1, 1], [1, 0, 1, 0], [2, 3, 0, -4], [-1, 2, 0, 0]])" "-%'MATRIXG6
#7&7&\"\"!F'\"\"\"F(7&F(F'F(F'7&\"\"#\"\"$F',$\"\"%!\"\"7&,$F(F/F+F'F'
" }{TEXT -1 33 "    with respect to the basis    " }{XPPEDIT 19 1 "[u[
1],u[2],u[3],u[4]]" "7&&%\"uG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }
{TEXT -1 1 " " }{TEXT 279 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 282 5 "( i )" }{TEXT -1 58 "    Use a transition
 matrix to calculate the matrix of    " }{XPPEDIT 19 1 "T" "I\"TG6\"" 
}{TEXT -1 34 "     with respect to the basis    " }{XPPEDIT 19 1 "[v[1
],v[2],v[3],v[4]]" "7&&%\"vG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }
{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT 283 8 "( ii )  " }{TEXT -1 28 "
 Using the generic vector   " }{XPPEDIT 19 1 "w = [a,b,c,d]" "/%\"wG7&
%\"aG%\"bG%\"cG%\"dG" }{TEXT -1 7 "  of   " }{XPPEDIT 19 1 "R^4" "*$%
\"RG\"\"%" }{TEXT -1 89 "    (written in standard Euclidean coordinate
s), verify that thetwo representations of   " }{XPPEDIT 19 1 "T" "I\"T
G6\"" }{TEXT -1 42 "  yield the same result when applied to   " }
{XPPEDIT 19 1 "w " "I\"wG6\"" }{TEXT -1 4 " .  " }}{PARA 0 "" 0 "" 
{TEXT 284 8 "( iii ) " }{TEXT -1 25 "What  is the matrix of   " }
{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 43 "   with respect to the stan
dard basis of   " }{XPPEDIT 19 1 "R^4" "*$%\"RG\"\"%" }{TEXT -1 1 "?" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 260 "" 0 "" {TEXT -1 11 "Exercise 3 " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Background worksheet:    \+
  " }{HYPERLNK 17 "transitionMatrixR4.mws" 1 "transitionMatrixR4.mws" 
"" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Let   " }{XPPEDIT 19 1 "V" "I\"VG6\"" }{TEXT -1 81 "  be t
he space of polynomials with real coefficients and degree 2 or less.  \+
Let  " }{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 5 " map " }{XPPEDIT 19 
1 "V" "I\"VG6\"" }{TEXT -1 5 "  to " }{XPPEDIT 19 1 "V" "I\"VG6\"" }
{TEXT -1 16 "  be defined by " }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}
{PARA 0 "" 0 "" {TEXT -1 16 "                " }{XPPEDIT 19 1 "p(x)  -
>exp(x)*diff(p(x)*exp(-x),x)" ":6#-%\"pG6#%\"xG7\"6$%)operatorG%&arrow
G6\"*&-%$expG6#F'\"\"\"-%%diffG6$*&-F%6#F'F1-F/6#,$F'!\"\"F1F'F1F,F," 
}{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 5 "( i )" }{TEXT -1 30 "    \+
  What is the matrix of   " }{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 
41 "   with respect to  the standard basis   " }{XPPEDIT 19 1 "[1,x,x^
2]" "7%\"\"\"%\"xG*$F$\"\"#" }{TEXT -1 4 "?  \n" }{TEXT 286 6 "( ii )
" }{TEXT -1 64 "     What is the transition matrix from the standard b
asis to   " }{XPPEDIT 19 1 "[x-1,x+1,x^2-1]" "7%,&%\"xG\"\"\"F%!\"\",&
F$F%F%F%,&*$F$\"\"#F%F%F&" }{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT 
287 8 "( iii  )" }{TEXT -1 28 "    What is the matrix of   " }
{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 33 "   with respect to  the  ba
sis   " }{XPPEDIT 19 1 "[x-1,x+1,x^2-1]" "7%,&%\"xG\"\"\"F%!\"\",&F$F%
F%F%,&*$F$\"\"#F%F%F&" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercise 4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Background worksheet:    \+
" }{HYPERLNK 17 "transitionMatrixR4.mws" 1 "transitionMatrixR4.mws" "
" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Let   " }{XPPEDIT 19 1 "V
" "I\"VG6\"" }{TEXT -1 81 "  be the space of polynomials with real coe
fficients and degree 3 or less.  Let  " }{XPPEDIT 19 1 "T" "I\"TG6\"" 
}{TEXT -1 5 " map " }{XPPEDIT 19 1 "V" "I\"VG6\"" }{TEXT -1 5 "  to " 
}{XPPEDIT 19 1 "V" "I\"VG6\"" }{TEXT -1 16 "  be defined by " }}{PARA 
0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 16 "          \+
      " }{XPPEDIT 19 1 "p(x)  ->  diff(p(x),x,x)-2*x*diff(p(x),x)+6*p(
x)  " ":6#-%\"pG6#%\"xG7\"6$%)operatorG%&arrowG6\",(-%%diffG6%-F%6#F'F
'F'\"\"\"*(\"\"#F3F'F3-F/6$-F%6#F'F'F3!\"\"*&\"\"'F3-F%6#F'F3F3F,F," }
{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 5 "( i )" }{TEXT -1 30 "    \+
  What is the matrix of   " }{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 
39 "   with respect to the standard basis  " }{XPPEDIT 19 1 "[1,x,x^2,
x^3]" "7&\"\"\"%\"xG*$F$\"\"#*$F$\"\"$" }{TEXT -1 1 "?" }}{PARA 0 "" 
0 "" {TEXT 289 6 "( ii )" }{TEXT -1 40 "     Directly calculate the ma
trix of   " }{XPPEDIT 19 1 "T" "I\"TG6\"" }{TEXT -1 30 "   with respec
t to the basis  " }{XPPEDIT 19 1 "[1,x,2*x^2-1,4*x^3-3*x]" "7&\"\"\"%
\"xG,&*&\"\"#F#*$F$F'F#F#F#!\"\",&*&\"\"%F#*$F$\"\"$F#F#*&F.F#F$F#F)" 
}{TEXT -1 38 "  (without using a transition matrix)." }}{PARA 0 "" 0 "
" {TEXT 290 8 "( iii  )" }{TEXT -1 88 "   By calculating a transition \+
matrix show explicitly that the two matrices are similar." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}{EXCHG 
{PARA 261 "" 0 "" {TEXT -1 50 "M309F03hw5R4.mws     A MapleV Release 4
 worksheet." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" 
{TEXT -1 42 "Author:  Brian E. Blank  (27 October 2003)" }}{PARA 264 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may \+
not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "incl
uding 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 more informati
on,  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 University in St. Louis
" }}{PARA 0 "" 0 "" {TEXT -1 26 "     St. Louis, MO   63130" }}{PARA 
0 "" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 33 "     Telepho
ne:    (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 2001 Brian E. Blank,  \+
All Rights Reserved." }}}}}{MARK "0 1 0" 4 }{VIEWOPTS 1 1 0 3 4 1802 }