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" {TEXT -1 47 "How to construct a random 5x5 matrix of rank 3?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "A := RandomMatrix(5,3).RandomMatrix(3,5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*kn*o9-%'MATRIXG6#7'7'\"%yB\"%*)))!
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{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rea
d \"c:/304/svd.txt\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "U,
 S, V := svd(A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6%%\"UG%\"SG%\"VG
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**4<>ZQcRCF4$!3cznCMkC:jF4Fdo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "N := V[1..-1,4..5];   # getting the column 4..5 from V" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'RTABLEG6%\"*oUHZ\"-%'MATRIXG6#7'7$$
!3Y534P:jF&*!#=$!3I<\"G(fP(*\\?F07$$!3GM5(\\#zBH5F0$\"3>0+O'yw*4hF07$$
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{MPLTEXT 1 0 29 "A.N;  # verify the null space" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\"*?AJZ\"-%'MATRIXG6#7'7$$!3SS+3%e#[\"f#!#H
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:= U[1..-1,4..5];  # getting left null space" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"*Wv_Y\"-%'MATRIXG6#7'7$$\"3al<06CBp
P!#=$!3fA6jp(GZs#!#>7$$!3m'=B%HxuhpF0$!3NDn#\\#p1@DF07$$\"3flJQZxx+BF0
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"" {MPLTEXT 1 0 15 "Transpose(A).M;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'RTABLEG6%\"*7!Gu9-%'MATRIXG6#7'7$$\"3i2R7c2-\"*G!#I$!3;>_ZMAJa:!#J
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60 "Question:  How to solve the homogeneous linear system  Ax=0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Answer:  \+
Get the SVD, from which we can get the null space. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Question:  How to solve g
eneral Ax=b?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 37 "Example:   Solve by LU decomposition." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Construct a testing problem" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "A := RandomMatrix(5,5)*1.0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
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LEG6%\"*gE8Y\"-%'MATRIXG6#7'7#\"\"$7#\"\"#7#!\"&7#\"\"!F-&%'VectorG6#%
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'RTABLEG6%\"*?G8Y\"-%'MATRIXG6
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{XPPMATH 20 "6#>6%%\"PG%\"LG%\"UG6%-%'RTABLEG6%\"*3r(y9-%'MATRIXG6#7'7
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