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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 27 "An introduction to Map
le's " }{TEXT 257 6 "linalg" }{TEXT 258 8 " package" }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Getting started" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "You need to enter the following command b
efore using the linear algebra package:" }{MPLTEXT 1 0 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 42 "Define a m by n matrix A with the command " }}{PARA 0 "" 
0 "" {TEXT -1 70 "A:=array(1..m,1..n,[entries of first row],[second ro
w],...,[mth row]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "A:=array(1..4
,1..5,[[12,-7,11,-9,5],[-9,4,-8,7,-3],[-6,11,-7,3,-9],[4,-6,10,-5,12]]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7&7'\"#7!\"(\"
#6!\"*\"\"&7'F-\"\"%!\")\"\"(!\"$7'!\"'F,F+\"\"$F-7'F0F5\"#5!\"&F*" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Row manipulation" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 66 "Use the command     addrow(A,R1,R2,t);   \+
 to replace R2 by R2+t R1" }}{PARA 0 "" 0 "" {TEXT -1 59 "If your inpu
t ends with a colon, the result is not printed." }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "A1:=addrow(A,1,2,9/12): A2:=addrow(
A1,1,3,1/2):\nA3:=addrow(A2,1,4,-1/3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#A3G-%'MATRIXG6#7&7'\"#7!\"(\"#6!\"*\"\"&7'\"\"!#!\"&\"\"%#\"
\"\"F3F4#\"\"$F37'F0#\"#:\"\"##!\"$F;F<#!#8F;7'F0#!#6F7#\"#>F7!\"##\"#
JF7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Use the command    mulrow(
A,R1,m);    to multiply row R1 of A by m" }{MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "A4:=mulrow(A3,1,1/12): A5:=mulrow(A4,2,-4
/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G-%'MATRIXG6#7&7'\"\"\"#!
\"(\"#7#\"#6F-#!\"$\"\"%#\"\"&F-7'\"\"!F*#!\"\"F4F7#F1F47'F6#\"#:\"\"#
#F1F=F>#!#8F=7'F6#!#6\"\"$#\"#>FD!\"##\"#JFD" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 91 "Use the command   pivot(A,i,j)   to row-manipulate the \+
matrix so that the entries above and" }}{PARA 0 "" 0 "" {TEXT -1 45 "b
elow the pivot position (i,j) are equal to 0" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "A6:=pivot(A5,2,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#A6G-%'MATRIXG6#7&7'\"\"\"\"\"!#\"\"%\"\"&#!#8\"#:#F*F17'F+F*#!\"\"
F.F4#!\"$F.7'F+F+F+F+!\"#7'F+F+#\"#GF.#!#TF1#\"$A\"F1" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 66 "Use the command   swaprow(A,R1,R2);    to
 swap rows R1 and R2 of A" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A7:=swaprow(A6,3,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#A7G-%'MATRIXG6#7&7'\"\"\"\"\"!#\"\"%\"\"&#!#8\"#:#F*F17'F+F*#!\"
\"F.F4#!\"$F.7'F+F+#\"#GF.#!#TF1#\"$A\"F17'F+F+F+F+!\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A8:=mulrow(A7,3,5/28):A9:=pivot(A8,
3,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A9G-%'MATRIXG6#7&7'\"\"\"
\"\"!F+#!#5\"#@#!#BF.7'F+F*F+#!#D\"#%)#!#8\"#U7'F+F+F*#!#TF4#\"#hF77'F
+F+F+F+!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A10:=mulrow(
A9,4,-1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$A10G-%'MATRIXG6#7&7'
\"\"\"\"\"!F+#!#5\"#@#!#BF.7'F+F*F+#!#D\"#%)#!#8\"#U7'F+F+F*#!#TF4#\"#
hF77'F+F+F+F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We finally obt
ain the row-reduced form of the matrix A." }{MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "A11:=pivot(A10,4,5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$A11G-%'MATRIXG6#7&7'\"\"\"\"\"!F+#!#5\"#@F+7'F+F*F+#
!#D\"#%)F+7'F+F+F*#!#TF2F+7'F+F+F+F+F*" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 44 "Row reduced form of a matrix and column span" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 83 "Maple has a routine which performs row-re
duction for you: use the command   rref(A)" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "R:=rref(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RG-%'MATRIXG6#7&7'\"\"\"\"\"!F+#!#5\"#@F+7'F+F*F+#!
#D\"#%)F+7'F+F+F*#!#TF2F+7'F+F+F+F+F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 95 "You can obtain vectors which span the column space of the
 matrix with the command   colspan(A)." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "colspan(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&-%'VECTORG6#7
&!\"(\"\"%\"#6!\"'-F%6#7&\"\"!F/\"#C!##*-F%6#7&F/F/F/\"#!)-F%6#7&F/\"
\"\"\"\")!#a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "The span of the \+
columns of A is the span of the pivot columns. If we remove the fourth
 column of A (using the command    delcols(A,4..4)), the span does not
 change." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "B:=d
elcols(A,4..4); colspan(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-
%'MATRIXG6#7&7&\"#7!\"(\"#6\"\"&7&!\"*\"\"%!\")!\"$7&!\"'F,F+F/7&F0F4
\"#5F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&-%'VECTORG6#7&!\"(\"\"%\"#
6!\"'-F%6#7&\"\"!\"\"\"\"\")!#a-F%6#7&F/F/\"#C!##*-F%6#7&F/F/F/!$o\"" 
}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "How to enter a matrix interac
tively" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "You can also enter a mat
rix interactively using the command    entermatrix(M)   after having" 
}}{PARA 0 "" 0 "" {TEXT -1 21 "defined the matrix M:" }{MPLTEXT 1 0 0 
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "M:=array(1..2,1..2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%&arrayG6%;\"\"\"\"\"#F(7\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "entermatrix(M);" }}}{EXCHG 
{PARA 0 "enter element 1,1 > " 0 "" {MPLTEXT 1 0 2 "1;" }}}{EXCHG 
{PARA 0 "enter element 1,2 > " 0 "" {MPLTEXT 1 0 2 "2;" }}}{EXCHG 
{PARA 0 "enter element 2,1 > " 0 "" {MPLTEXT 1 0 2 "3;" }}}{EXCHG 
{PARA 0 "enter element 2,2 > " 0 "" {MPLTEXT 1 0 2 "4;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7$\"\"\"\"\"#7$\"\"$\"\"%" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Solution of a matrix equation" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "b:=vector([1,2,3,4]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'VECTORG6#7&\"\"\"\"\"#\"\"$\"
\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Use the command   linsolve
(A,b)   to solve the linear system " }{XPPEDIT 18 0 "A x = b" "/*&%\"A
G\"\"\"%\"xGF%%\"bG" }{TEXT -1 46 ". The free variables, if any, are i
n the form " }{XPPEDIT 18 0 "_t[1], _t[2]" "6$&%#_tG6#\"\"\"&F$6#\"\"#
" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "_t[k]" "&%#_tG6#%\"kG" }{TEXT 
-1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sol:=linsolve(A,b);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG-%'VECTORG6#7'&%#_tG6#\"\"\",&F
)#\"\"&\"\")#\"\"(\"\"%F,,&F)#\"#T\"#S#\"$6&\"#?F,,&F)#\"#@\"#5#\"$@\"
F/F,!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "To find a particular s
olution, set the free variable(s) to 0" }{MPLTEXT 1 0 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "subs(_t[1]=0,linsolve(A,b));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'VECTORG6#7'\"\"!#\"\"(\"\"%#\"$6&\"#?#\"$@\"\"
\"&!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 286 "To find the \"vector a
ssociated with each free variable\", substract the particular solution
 from the general solution and then set the corresponding free variabl
e to one and all of the others (if any) to 0. You can use the command \+
  matadd(v,w,c1,c2)   to compute the vector  c1 v + c2 w" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 69 "subs(_t[1]=1,matadd(linsolve(A,b),subs(_t[1]=
0,linsolve(A,b)),1,-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG
6#7'\"\"\"#\"\"&\"\")#\"#T\"#S#\"#@\"#5\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 112 "This is better achieved by finding the verctors in the
 kernel (or null space) of A, with the command   kernel(A)" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "kernel(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#<#-%'VECTORG6#7'\"\"\"#\"\"&\"\")#\"#T\"#S#\"#@\"#5\"\"!" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Eigenvalues, eigenvectors and Jor
dan form" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Consider the following
 matrix" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A:=ar
ray(1..3,1..3,[[11,-2,-1],[0,12,-4],[1,2,5]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"#6!\"#!\"\"7%\"\"!\"#7!\"%7%\"
\"\"\"\"#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "To find its eig
envalues, we can use the command    charpoly(A,lambda)   and factor th
e result" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fact
or(charpoly(A,lambda));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%'lambd
aG\"\"\"!#7F&F&,&F%F&!\")F&\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
44 "We can also use the command   eigenvalues(A)" }{MPLTEXT 1 0 0 "" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvalues(A);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"#7\"\")F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "
The command  eigenvectors(A)  returns lists showing each eigenvalue, i
ts multiplicity, and the corresponding eigenvector(s)" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$7%\"#7\"\"\"<#-%'VECTORG6#7%!\"#F%\"\"!7%\"\")\"\"#<#-F(6#7%F%F%
F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "We now reduce A to its Jor
dan form, first by finding a generalized eigenvector. We then solve th
e equation  " }{XPPEDIT 18 0 "v[2] = (A - lambda* I )*v[3]" "/&%\"vG6#
\"\"#*&,&%\"AG\"\"\"*&%'lambdaGF*%\"IGF*!\"\"F*&F$6#\"\"$F*" }{TEXT 
-1 8 ", where " }{XPPEDIT 18 0 "v[2]=[1,1,1]" "/&%\"vG6#\"\"#7%\"\"\"
\"\"\"\"\"\"" }{TEXT -1 22 " is an eigenvector of " }{XPPEDIT 18 0 "A
" "I\"AG6\"" }{TEXT -1 19 " with eigenvalue 8." }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "I3:=array(1..3,1..3,identity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I3G-%&arrayG6&%)identityG;\"\"\"\"
\"$F)7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v[2]=subs(_t[1]
=0,linsolve(A-8*I3,[1,1,1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"
vG6#\"\"#-%'VECTORG6#7%#\"\"\"F'#F-\"\"%\"\"!" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 36 "We can then define the Jordan form J" }{MPLTEXT 1 0 0 
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "J:=array(1..3,1..3,[[12,0,0],[
0,8,1],[0,0,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'MATRIXG6
#7%7%\"#7\"\"!F+7%F+\"\")\"\"\"7%F+F+F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "and the matrix " }{XPPEDIT 18 0 "M" "I\"MG6\"" }{TEXT -1 
19 " whose columns are " }{XPPEDIT 18 0 "v[1]" "&%\"vG6#\"\"\"" }
{TEXT -1 20 ", an eigenvector of " }{XPPEDIT 18 0 "A" "I\"AG6\"" }
{TEXT -1 21 " with eigenvalue 12; " }{XPPEDIT 18 0 "v[2]" "&%\"vG6#\"
\"#" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "v[3]" "&%\"vG6#\"\"$" }
{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "M:=array(1..3,1..3
,[[-2,1,1/2],[1,1,1/4],[0,1,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"MG-%'MATRIXG6#7%7%!\"#\"\"\"#F+\"\"#7%F+F+#F+\"\"%7%\"\"!F+F2" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "To check that " }{XPPEDIT 18 0 "A \+
= M*J*M^(-1)" "/%\"AG*(%\"MG\"\"\"%\"JGF&)F%,$\"\"\"!\"\"F&" }{TEXT 
-1 28 ", we compute the inverse of " }{XPPEDIT 18 0 "M" "I\"MG6\"" }
{TEXT -1 2 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(M);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#!\"\"\"\"%#\"\"\"\"\"
#F(7%\"\"!F/F,7%F,F-!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "evalu
ate the product of the 3 matrices" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "C:=evalm(M&*J&*inverse(M));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7%7%\"#6!\"#!\"\"7%\"\"!\"#7!\"%7%\"
\"\"\"\"#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "and substract t
he result from the matrix " }{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 2 
". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A-C);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"!F(F(F'F'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 48 "A more direct way of obtaining a Jordan form of " 
}{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 67 " is to use the command  jo
rdan(A,P), which returns the Jordan form " }{XPPEDIT 18 0 "J" "I\"JG6
\"" }{TEXT -1 32 " of A, together with the matrix " }{XPPEDIT 18 0 "P
" "I\"PG6\"" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "J =inverse(P)*
A*P" "/%\"JG*(-%(inverseG6#%\"PG\"\"\"%\"AGF)F(F)" }{TEXT -1 1 "." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "jordan(A,P);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'MATRIXG6#7%7%\"#7\"\"!F)7%F)\"\")\"\"\"7%F)F)F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(P);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#\"\"\"\"\"#F)F(7%#!\"\"\"\"%F)#F)F.7
%\"\"!F)F1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "
6 0 0" 32 }{VIEWOPTS 1 1 0 1 1 1803 }