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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Least Squares" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "First we find the least squares so
lution of Ax = b for a given matrix A and vector b." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 75 "A := Matrix([[1,1,0,0],[1,1,0,0],[1,0,1,0
],[1,0,1,0],[1,0,0,1],[1,0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"AG-%'RTABLEG6%\")74)\\\"-%'MATRIXG6#7(7&\"\"\"F.\"\"!F/F-7&F.F/F.F
/F07&F.F/F/F.F1%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"b := Vector[column]([-3,-1,0,2,5,1]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"bG-%'RTABLEG6%\")%3Kh\"-%'MATRIXG6#7(7#!\"$7#!\"\"7#\"\"!7#
\"\"#7#\"\"&7#\"\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "Transpose(A).A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6%\")[_)\\\"-%'MATRIXG6#7&7&\"\"'\"\"#F-F-7&F-F-\"\"!F/7&F-F
/F-F/7&F-F/F/F-%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"Transpose(A).b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")/A8
;-%'MATRIXG6#7&7#\"\"%7#!\"%7#\"\"#7#\"\"'&%'VectorG6#%'columnG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "aug:=Matrix([Transpose(A).A,
Transpose(A).b]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$augG-%'RTABLEG
6%\")c$yh\"-%'MATRIXG6#7&7'\"\"'\"\"#F/F/\"\"%7'F/F/\"\"!F2!\"%7'F/F2F
/F2F/7'F/F2F2F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "ReducedRowEchelonForm(aug);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
RTABLEG6%\")'p!=;-%'MATRIXG6#7&7'\"\"\"\"\"!F-F,\"\"$7'F-F,F-!\"\"!\"&
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-1 23 "The general solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 12 "x1 = 3 - x4," }}{PARA 256 "" 0 "" 
{TEXT -1 13 "x2 = -5 + x4," }}{PARA 256 "" 0 "" {TEXT -1 13 "x3 = -2 +
 x4," }}{PARA 256 "" 0 "" {TEXT -1 10 "x4 = free." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The general least squares
 solution of Ax = b has the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 40 "x = [3, -5, -2, 0] + x4 * [-1, 1, 1, 1]
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Next we find the equation of the \+
least squares line that best fits the data points (2,1), (5,2), (7,3),
 and (8,3)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "B := Matrix(
[[1,2],[1,5],[1,7],[1,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-
%'RTABLEG6%\")%3tU\"-%'MATRIXG6#7&7$\"\"\"\"\"#7$F.\"\"&7$F.\"\"(7$F.
\"\")%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "y := Vect
or[column]([1,2,3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'RTA
BLEG6%\")CB8;-%'MATRIXG6#7&7#\"\"\"7#\"\"#7#\"\"$F1&%'VectorG6#%'colum
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).y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&augByG-%'RTABLEG6%\")))y&
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{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"))Q'>;-%'MATRIXG6#7$7%\"
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"" {TEXT -1 75 "Next we generate supressed plots for the points and th
e least squares line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "li
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