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{SECT 0 {EXCHG {PARA 215 "" 0 "" {TEXT 220 23 "Math 2250-2 Summer 2007
" }{TEXT 220 39 "\nLinear Algebra Computations with Maple" }}{PARA 
216 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 
216 9 "restart; " }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 49 "wit
h(linalg):   # load the linear algebra library" }}{PARA 7 "" 1 "" 
{TEXT -1 80 "Warning, the protected names norm and trace have been red
efined and unprotected\n" }}}{EXCHG {PARA 219 "" 0 "" {TEXT 214 29 "De
fining matrices and vectors" }}{PARA 220 "" 0 "" {TEXT -1 119 "Use the
 Matrix command. Define it row by row with each row inside [.] and not
e the number of brackets you need to put. " }{TEXT -1 49 "\nAlso, use \+
upper case calling the Matrix function" }}}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 216 41 "A := Matrix([ [1,2,3],[4,5,6],[7,8,9] ]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"(/fN%-%'MATRIXG6#7%7%\"
\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*%'MatrixG" }}}{EXCHG 
{PARA 217 "> " 0 "" {MPLTEXT 1 216 41 "B := Matrix([ [1,2,0],[0,5,2],[
2,8,0] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\"(CDV
%-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"!7%F0\"\"&F/7%F/\"\")F0%'MatrixG" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 29 "v := Matrix([ [1],[4],[3
] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'RTABLEG6%\"(GtV%-%'M
ATRIXG6#7%7#\"\"\"7#\"\"%7#\"\"$%'MatrixG" }}}{EXCHG {PARA 220 "" 0 "
" {TEXT -1 128 "You can also set-up entries of a matrix automatically.
 For example, define the (i,j) entry of a matrix by the following func
tion" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 20 "f := (i,j) -> (
i+j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"iG%\"jG6\"6$%)op
eratorG%&arrowGF),&9$\"\"\"9%F/F)F)F)" }}}{EXCHG {PARA 220 "" 0 "" 
{TEXT -1 65 "Then you can set up an 8x8 matrix with entries as above b
y typing" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 17 "G := Matrix
(8,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'RTABLEG6%\"(?JX%-%'
MATRIXG6#7*7*\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*7*F/F0F1F2F3F4F5
\"#57*F0F1F2F3F4F5F7\"#67*F1F2F3F4F5F7F9\"#77*F2F3F4F5F7F9F;\"#87*F3F4
F5F7F9F;F=\"#97*F4F5F7F9F;F=F?\"#:7*F5F7F9F;F=F?FA\"#;%'MatrixG" }}}
{EXCHG {PARA 220 "" 0 "" {TEXT -1 74 "So that the (1,1) entry is 1+1 =
 2 and the (1,2) entry is 1+2=3 and so on." }}}{EXCHG {PARA 222 "" 0 "
" {TEXT 213 17 "Matrix operations" }}}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 216 31 "A+B;      # adding two matrices" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6%\"(C.k%-%'MATRIXG6#7%7%\"\"#\"\"%\"\"$7%F
-\"#5\"\")7%\"\"*\"#;F3%'MatrixG" }}}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 216 45 "A.B;      # matrix multiply using '.' not '*'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(C!*o%-%'MATRIXG6#7%7%\"
\"(\"#O\"\"%7%\"#;\"#\")\"#57%\"#D\"$E\"F0%'MatrixG" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 216 4 "A*B;" }}{PARA 8 "" 1 "" {TEXT -1 45 "E
rror, (in rtable/Product) invalid arguments\n" }}}{EXCHG {PARA 217 "> \+
" 0 "" {MPLTEXT 1 216 42 "A.v;    # matrix and vector multiplication" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(+fr%-%'MATRIXG6#7%7#
\"#=7#\"#U7#\"#m%'MatrixG" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 
216 53 "A^2;    # you can take the square of a matrix so that" }
{MPLTEXT 1 216 20 "\n        # A^2 = A*A" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'RTABLEG6%\"(G&\\Z-%'MATRIXG6#7%7%\"#I\"#O\"#U7%\"#m\"#\")\"#'
*7%\"$-\"\"$E\"\"$]\"%'MatrixG" }}}{EXCHG {PARA 217 "> " 0 "" 
{MPLTEXT 1 216 4 "A*A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%
\"(wgx%-%'MATRIXG6#7%7%\"#I\"#O\"#U7%\"#m\"#\")\"#'*7%\"$-\"\"$E\"\"$]
\"%'MatrixG" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 57 "(A^4);  \+
 # you can raise the matrix to any power you want" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\"('R<[-%'MATRIXG6#7%7%\"%gv\"%)G*\"&;5\"7%
\"&=r\"\"&L5#\"&[\\#7%\"&wm#\"&yF$\"&!))Q%'MatrixG" }}}{EXCHG {PARA 
224 "" 0 "" {TEXT 211 31 "Identity Matrix and Zero Matrix" }}}{EXCHG 
{PARA 217 "> " 0 "" {MPLTEXT 1 216 58 "I3 := Matrix(3,3,shape=identity
);    # 3x3 identity matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I3G-
%'RTABLEG6%\"(!)f$[-%'MATRIXG6#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/F.%'Matr
ixG" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 49 "Z3 := Matrix(3,3
);              # 3x3 zero matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#Z3G-%'RTABLEG6%\"(s3%[-%'MATRIXG6#7%7%\"\"!F.F.F-F-%'MatrixG" }}}
{EXCHG {PARA 220 "" 0 "" {TEXT 221 14 "Row Operations" }{TEXT -1 39 "
\nYou need to know three function calls:" }{TEXT -1 9 "\n* mulrow" }
{TEXT -1 10 "\n* swaprow" }{TEXT -1 9 "\n* addrow" }{TEXT -1 52 "\nNot
e all of these functions start with a lower case" }}}{EXCHG {PARA 217 
"> " 0 "" {MPLTEXT 1 216 52 "M := Matrix([ [2,2,-1,-6],[3,-3,5,36],[5,
4,2,13] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\"(3d
%[-%'MATRIXG6#7%7&\"\"#F.!\"\"!\"'7&\"\"$!\"$\"\"&\"#O7&F4\"\"%F.\"#8%
'MatrixG" }}}{EXCHG {PARA 220 "" 0 "" {TEXT -1 54 "The function mulrow
 will multiply a row by a constant " }{TEXT -1 57 "\n      mulrow(A,2,
4) will multiply row 2 of matrix A by 4" }{TEXT -1 22 "\nAnother examp
le below" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 64 "M1 := mulro
w(M, 1 , 1/2);    # multiply row 1 of matrix M by 1/2" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#M1G-%'matrixG6#7%7&\"\"\"F*#!\"\"\"\"#!\"$7&\"
\"$F.\"\"&\"#O7&F1\"\"%F-\"#8" }}}{EXCHG {PARA 220 "" 0 "" {TEXT -1 
70 "The function addrow will multiply a row and then add it to another
 row" }{TEXT -1 80 "\n  addrow(A,1,3,5) will multiply row 1 of matrix \+
A by 5 and then add it to row 3" }{TEXT -1 4 "\nSo " }{TEXT -1 47 "\n \+
    addrow(matrix_name,  row_a, row_b, coeff)" }}{PARA 220 "" 0 "" 
{TEXT -1 6 "means " }{TEXT -1 25 "\n     coeff*row_a + row_b" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 47 "M2 := addrow(M1,1,2,-3);
   # -3*row(1) + row(2)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'ma
trixG6#7%7&\"\"\"F*#!\"\"\"\"#!\"$7&\"\"!!\"'#\"#8F-\"#X7&\"\"&\"\"%F-
F3" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 46 "M3 := addrow(M2,1
,3,-5);  # -5*row(1) + row(3)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3
G-%'matrixG6#7%7&\"\"\"F*#!\"\"\"\"#!\"$7&\"\"!!\"'#\"#8F-\"#X7&F0F,#
\"\"*F-\"#G" }}}{EXCHG {PARA 220 "" 0 "" {TEXT -1 40 "Finally, swaprow
 will just swap two rows" }{TEXT -1 17 "\n  swaprow(A,1,3)" }}{PARA 
220 "" 0 "" {TEXT -1 37 "will swap row 1 and row 3 of matrix A" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 43 "M4 := swaprow(M3,2,3);  \+
 # swap row 2 and 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M4G-%'matrix
G6#7%7&\"\"\"F*#!\"\"\"\"#!\"$7&\"\"!F,#\"\"*F-\"#G7&F0!\"'#\"#8F-\"#X
" }}}{EXCHG {PARA 225 "" 0 "" {TEXT 207 49 "Augmenting matrices and re
duced row echelon form " }}}{EXCHG {PARA 226 "" 0 "" {TEXT 200 76 "The
re is an automatic way to reduce a matrix to the reduced row echelon f
orm" }}}{EXCHG {PARA 227 "> " 0 "" {MPLTEXT 1 0 16 "RREFM:= rref(M);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RREFMG-%'matrixG6#7%7&\"\"\"\"\"!
F+F*7&F+F*F+!\"\"7&F+F+F*\"\"'" }}}{EXCHG {PARA 226 "" 0 "" {TEXT 200 
32 "To augment two matrices together" }}}{EXCHG {PARA 227 "> " 0 "" 
{MPLTEXT 1 0 19 "Av:= Matrix([A,v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#AvG-%'RTABLEG6%\"(s30&-%'MATRIXG6#7%7&\"\"\"\"\"#\"\"$F.7&\"\"%\"
\"&\"\"'F27&\"\"(\"\")\"\"*F0%'MatrixG" }}}{EXCHG {PARA 227 "> " 0 "" 
{MPLTEXT 1 0 19 "AB:= Matrix([A,B]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#ABG-%'RTABLEG6%\"([s(Q-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F.F/\"\"!7(
\"\"%\"\"&\"\"'F1F4F/7(\"\"(\"\")\"\"*F/F8F1%'MatrixG" }}}{EXCHG 
{PARA 228 "" 0 "" {TEXT 206 33 "Computing determinant and inverse" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 61 "B_inv := Matrix(inverse(
B));   # take the inverse of matrix B" }{MPLTEXT 1 216 98 "\n         \+
                      # you need to set this as a Matrix in order to d
o operations after" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&B_invG-%'RTAB
LEG6%\"(O$z^-%'MATRIXG6#7%7%\"\"#\"\"!#!\"\"F.7%F0F/#\"\"\"\"\"%7%#\"
\"&F5#F4F.#!\"&\"\")%'MatrixG" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 
1 216 83 "inverse(B).v;    # this is what happens if you don't assign \+
the inverse as a matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\".G6$-%'
matrixG6#7%7%\"\"#\"\"!#!\"\"F+7%F-F,#\"\"\"\"\"%7%#\"\"&F2#F1F+#!\"&
\"\")-%'RTABLEG6%\"(GtV%-%'MATRIXG6#7%7#F17#F27#\"\"$%'MatrixG" }}}
{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 21 "Matrix(inverse(B)).v;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(#4D`-%'MATRIXG6#7%7##
\"\"\"\"\"#7##F-\"\"%7##\"#6\"\")%'MatrixG" }}}{EXCHG {PARA 217 "> " 
0 "" {MPLTEXT 1 216 81 "A_inv := Matrix(inverse(A));   # Maple will te
ll you if the inverse doesn't exist" }{MPLTEXT 1 216 64 "\n           \+
                    # our matrix A above is singular" }}{PARA 8 "" 1 "
" {TEXT -1 36 "Error, (in inverse) singular matrix\n" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 216 83 "Matrix(B^(-1));        # you can also
 compute inverse by raising to the power of -1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6%\"(7PV&-%'MATRIXG6#7%7%\"\"#\"\"!#!\"\"F,7%
F.F-#\"\"\"\"\"%7%#\"\"&F3#F2F,#!\"&\"\")%'MatrixG" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 216 47 "det(B);   # compute the determinant u
sing det()" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\")" }}}{EXCHG {PARA 
217 "> " 0 "" {MPLTEXT 1 216 89 "det(A);   # this confirms that indeed
 A inverse does not exist as the determinant is zero" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 210 28 "Eig
envalues and Eigenvectors" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 
216 59 "e := eigenvals(A);    # compute the eigenvalues of matrix A" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG6%\"\"!,&#\"#:\"\"#\"\"\"*(\"\"
$F+F*!\"\"\"#L#F+F*F+,&F(F+*(F-F+F*F.F/F0F." }}}{EXCHG {PARA 220 "" 0 
"" {TEXT -1 100 "Note that one of the eigenvalue is zero which again s
hows that matrix A is singular (not invertible)" }}}{EXCHG {PARA 217 "
> " 0 "" {MPLTEXT 1 216 41 "evalf(eigenvals(A));   # numerical values
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"!F$$\"+(R%o6;!\")$!+qR%o6\"!
\"*" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 42 "v := [eigenvecto
rs(A)];   # eigenvectors  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%
7%\"\"!\"\"\"<#-%'vectorG6#7%F(!\"#F(7%,&#\"#:\"\"#F(*(\"\"$F(F3!\"\"
\"#L#F(F3F(F(<#-F+6#7%,&#F(F3F6*(F5F(\"#AF6F7F8F(,&#F(\"\"%F(*(F5F(\"#
WF6F7F8F(F(7%,&F1F(*(F5F(F3F6F7F8F6F(<#-F+6#7%,&#F(F3F6*(F5F(F@F6F7F8F
6,&FBF(*(F5F(FEF6F7F8F6F(" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 
216 33 "v[1][1];   # the first eigenvalue" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 48 
"v[1][2];   # the multiplicity of that eigenvalue" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 216 
42 "v[1][3];   # the corresponding eigenvector" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#-%'vectorG6#7%\"\"\"!\"#F(" }}}{EXCHG {PARA 217 "> " 
0 "" {MPLTEXT 1 216 76 "v[2][1]; v[2][3];  # the second eigenvalue and
 its corresponding eigenvector" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#
\"#:\"\"#\"\"\"*(\"\"$F'F&!\"\"\"#L#F'F&F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#-%'vectorG6#7%,&#\"\"\"\"\"#!\"\"*(\"\"$F*\"#AF,\"#L#
F*F+F*,&#F*\"\"%F**(F.F*\"#WF,F0F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 230 "" 0 "" {TEXT -1 0 "" }}}{PARA 
231 "" 0 "" {TEXT -1 0 "" }}{PARA 231 "" 0 "" {TEXT -1 0 "" }}{PARA 
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