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"Helvetica" 1 14 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 "Helvetica" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 207 26 "Linear Algebra Package in " }{TEXT 208 5 "Maple" }{TEXT 207 23 ": MATH6050 Optimization" }}} {EXCHG {PARA 214 "" 0 "" {TEXT 221 0 "" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 29 "restart;with(LinearAlgebra): " }{TEXT 207 30 "Loading linear algebra package" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " ? LinearAlgebra;" }}}{SECT 0 {PARA 241 "" 0 "" {TEXT 222 48 " Vectors \+ and elementary operations with vectors." }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 0 39 "u:=Vector([2,-1,2]);v:=Vector([1,2,4]);" }{TEXT 223 32 " Entering vector-columns u and v" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 14 "Dimension(u); " }{TEXT 224 25 "Dimension of the vecto r u" }}{PARA 215 "" 0 "" {TEXT 225 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 6 "u[3]; " }{TEXT 226 84 "Vectors are one-dimensional arr ays, here we evaluate the third component of vector u" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 0 5 "u+v; " }{TEXT 227 137 "Sum of two \+ vectors u and v (note that Maple likes to operate in symbolic mode and you need to tell Maple that you want a numerical answer" }{MPLTEXT 1 0 2 " " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 0 17 "(u+v); (3*u-2* v);" }{TEXT 228 13 " Use command " }{TEXT 209 7 "evalm " }{TEXT 228 22 "(evaluate to a matrix)" }{TEXT 210 1 " " }{TEXT 228 61 "to obtain \+ a numerical answer for vector and matrix operations" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 0 17 "DotProduct(u,v); " }{TEXT 229 45 " Comp uting a dot product of vectors u and v" }}}{EXCHG {PARA 206 "> " 0 " " {MPLTEXT 1 0 11 "Norm(u,2); " }{TEXT 230 45 "Computing a norm (or l ength) of the vector u" }}}{EXCHG {PARA 252 "> " 0 "" {MPLTEXT 1 0 47 "VectorAngle(u,v); evalf(VectorAngle(u,v),100); " }{TEXT 231 145 "Comp uting an angle between vectors u and v. Note that in order to obtain a decimal approximation of the numerical answer you need to use command " }{TEXT 211 5 "evalf" }}}}{SECT 0 {PARA 255 "" 0 "" {TEXT 232 48 "Ma trices and elementary operations with matrices" }}{PARA 228 "" 0 "" {TEXT 233 0 "" }}{PARA 227 "" 0 "" {TEXT 234 0 "" }}{EXCHG {PARA 248 " > " 0 "" {MPLTEXT 1 0 39 "A:=Matrix([[2,3,1],[4,5,6],[2,3,1]]); " } {TEXT 235 1 " " }{TEXT 207 1 " " }{TEXT 235 54 "Entering the matrix A \+ which is two-dimensional array " }}}{EXCHG {PARA 259 "> " 0 "" {MPLTEXT 1 0 87 "A[2,3]; a1:=Column(A,1); a2:=Column(A,2); a3:=Column( A,3); b1:=Row(A,1); b2:=Row(A,2); " }{TEXT 212 126 "Here we evaluate ( 2,3)-entry of matrix A and the first , the second and the third vector -column of A and some rows of matrix A" }}}{EXCHG {PARA 249 "> " 0 "" {MPLTEXT 1 0 60 "A1:=Matrix([a3,a1]); A2:=Matrix([a2,a1]); B:=Transpos e(A1); " }{TEXT 236 24 "Here we use the command " }{TEXT 213 6 "Matrix " }{TEXT 207 1 " " }{TEXT 236 124 "to produce matrix A1 which columns \+ are a3 and a1 respectively and A2 with columns a2 and a1, matrix B wi th rows b2 and b1) " }}}{EXCHG {PARA 245 "> " 0 "" {MPLTEXT 1 0 21 "(A 1+A2);(3*A1-2*A2); " }{TEXT 237 71 "Computing a sum of matrices A1 and A2 and also their linear combination" }}{PARA 224 "" 0 "" {TEXT 238 0 "" }}}{EXCHG {PARA 246 "" 0 "" {TEXT 239 7 " " }}}{EXCHG {PARA 247 "> " 0 "" {MPLTEXT 1 0 16 "A.A2; B.A; A.u; " }{TEXT 240 25 " Computing matrix products" }}{PARA 223 "" 0 "" {TEXT 241 0 "" }}} {EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 27 "M:=Matrix(2,3,[2,3,-1,2]); " }{TEXT 242 76 "Another way to enter matrices is to use the command \+ dimensions of the matrix" }}{PARA 244 "" 0 "" {TEXT 243 0 "" }}{PARA 243 "" 0 "" {TEXT 244 0 "" }}}{EXCHG {PARA 238 "> " 0 "" {MPLTEXT 1 0 33 "f:=(i,j)->i+j;N:=Matrix(2,5,f); " }{TEXT 245 82 "Useful way of en tering a matrix when you know how its (i,j)-entry depends on (i,j)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 214 35 "f:=(i,j)->x^(i+j); N1:=Ma trix(4,f);" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 0 14 "Transpose(N ); " }{TEXT 246 35 "Computing the transpose of matrix M" }}{PARA 239 " " 0 "" {TEXT 247 0 "" }}}{EXCHG {PARA 235 "> " 0 "" {MPLTEXT 1 0 19 "D iagonalMatrix(u); " }{TEXT 207 54 "Entering the diagonal matrix corres ponding to vector u" }}}}{SECT 0 {PARA 217 "" 0 "" {TEXT 248 34 "Speci al operations with matrices: " }{TEXT 215 24 "rref, rank, det, inverse " }}{PARA 256 "" 0 "" {TEXT 249 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 0 36 "M:=Matrix(6,8,(i,j)->i+j); Rank(M); " }{TEXT 250 15 " Use command " }{TEXT 216 5 "rank " }{TEXT 250 16 "to find its rank" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 0 60 "N:=RandomMatrix(5,5); \+ Determinant(N); NI:=MatrixInverse(N); " }{TEXT 251 84 "For randomly ge nerated matrix N we compute its determinant and the inverse matrix NI" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 14 "Map(evalf,NI);" } {MPLTEXT 1 252 2 " " }{MPLTEXT 1 252 1 " " }{MPLTEXT 1 252 1 " " }}} {EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 0 5 "NI.N;" }{TEXT 253 37 " We c heck that NI is the inverse of N" }}}}{SECT 0 {PARA 216 "" 0 "" {TEXT 254 34 "Linear systems and their solutions" }}{PARA 251 "" 0 "" {TEXT 255 0 "" }}{EXCHG {PARA 250 "> " 0 "" {MPLTEXT 1 0 43 "A:=RandomMatrix (4,4); b:=RandomMatrix(4,1);" }}}{EXCHG {PARA 222 "> " 0 "" {MPLTEXT 1 0 20 "x:=LinearSolve(A,b);" }{TEXT 256 24 " To solve linear system " }{TEXT 217 5 "Ax=b " }{TEXT 256 11 "use command" }{TEXT 218 11 " lins olve. " }{TEXT 256 88 "You can also find general solution for a rectan gular matrix A as it is illustrated below" }}{PARA 218 "" 0 "" {TEXT 257 0 "" }}}{EXCHG {PARA 232 "> " 0 "" {MPLTEXT 1 0 54 "A:=Matrix(5,8, (i,j)->i/(i+j)); b:=Matrix(5,1,i->i^2);\n" }{MPLTEXT 1 0 21 "x:=Linear Solve(A,b); " }{TEXT 258 103 "Note that rank(A)=5 which implies that a general solution depends on 3=n-rank(A) parameters t_1,t_2,t_3" }}} {EXCHG {PARA 231 "> " 0 "" {MPLTEXT 1 0 52 "A:=Matrix(6,4,(i,j)->j/(i+ j)); b:=Vector(6,i->i^2); " }{TEXT 259 30 "Entering matrix A and vecto r b" }}{PARA 229 "> " 0 "" {MPLTEXT 1 0 37 "M1:=Matrix([A,b]); Rank(M1 );Rank(A); " }{TEXT 260 32 "Checking that the linear system " }{TEXT 219 4 "Ax=b" }{TEXT 260 47 " is inconsistent by computing ranks of mat rices" }}{PARA 233 "> " 0 "" {MPLTEXT 1 0 23 "x:=LeastSquares(A,b); " }{TEXT 261 51 "Computing the least-squares solution of this system" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 220 11 "Eigenvalues" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=Matrix(3,3,(i,j)->1/(i+j));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Map(evalf,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "C onditionNumber(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 125 100 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }