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"Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 20 "Inner Product Spaces" }} {PARA 19 "" 0 "" {TEXT 257 14 "Brian E. Blank" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Maple Functions (Builtins)\n" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "allvalues" 2 "allvalues" "" }{TEXT -1 6 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "dotprod" 2 " linalg[dotprod]" "" }{TEXT -1 6 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "GramSchmidt" 2 "linalg[ GramSchmidt ]" "" }{TEXT -1 8 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "inner prod" 2 "linalg[innerprod]" "" }{TEXT -1 6 " " }}{PARA 15 "" 0 " " {TEXT -1 1 " " }{HYPERLNK 17 "mtaylor" 2 "mtaylor" "" }{TEXT -1 4 " \+ " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "rand" 2 "rand" " " }{TEXT -1 3 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "r eadlib" 2 "readlib" "" }{TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "randmatrix" 2 "linalg[randmatrix]" "" }{TEXT -1 4 " " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "transpose" 2 "l inalg[transpose]" "" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Maple Functions (Homebrew s)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 261 12 "bilinearForm" }{TEXT -1 15 " Section 3 " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 258 12 "skewSymmPart" }{TEXT -1 12 " \+ Section 2 " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 259 10 "symmPart " }{TEXT -1 15 "Section 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "0. Introduction" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "This worksheet discusse s inner product spaces. \n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "1 . The Transpose of a Matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 319 11 "Definition:" } {TEXT -1 8 " Let " }{XPPEDIT 320 1 "M^(m,n)" ")%\"MG6$%\"mG%\"nG" } {TEXT -1 19 " (respectively " }{XPPEDIT 322 1 "M^(n,m)" ")%\"MG6$% \"nG%\"mG" }{TEXT -1 21 ") be the space of " }{XPPEDIT 321 1 "m*` x `*n " "*(%\"mG\"\"\"%$~x~GF$%\"nGF$" }{TEXT -1 21 " (respectively \+ " }{XPPEDIT 323 1 "n*` x `*m" "*(%\"nG\"\"\"%$~x~GF$%\"mGF$" } {TEXT -1 20 ") matrices. If " }{XPPEDIT 325 1 "A" "I\"AG6\"" } {TEXT -1 10 " is an " }{XPPEDIT 326 1 "m*` x `*n " "*(%\"mG\"\"\"%$ ~x~GF$%\"nGF$" }{TEXT -1 29 " matrix then we obtain an " }{XPPEDIT 327 1 "n*` x `*m" "*(%\"nG\"\"\"%$~x~GF$%\"mGF$" }{TEXT -1 12 " matr ix " }{XPPEDIT 328 1 "A^t" ")%\"AG%\"tG" }{TEXT -1 15 ", called the \+ " }{TEXT 331 9 "transpose" }{TEXT -1 7 " of " }{XPPEDIT 330 1 "A " "I\"AG6\"" }{TEXT -1 33 ", by writing the first row of " } {XPPEDIT 332 1 "A" "I\"AG6\"" }{TEXT -1 27 " as the first column of \+ " }{XPPEDIT 333 1 "A^t" ")%\"AG%\"tG" }{TEXT -1 24 " , the second row of " }{XPPEDIT 334 1 "A" "I\"AG6\"" }{TEXT -1 28 " as the second \+ column of " }{XPPEDIT 335 1 "A^t" ")%\"AG%\"tG" }{TEXT -1 12 " , and so on" }{TEXT 336 1 "." }{TEXT -1 14 " Thus, the " }{XPPEDIT 329 1 "i,j^th" "6$%\"iG)%\"jG%#thG" }{TEXT -1 14 " entry of " } {XPPEDIT 337 1 "A^t" ")%\"AG%\"tG" }{TEXT -1 13 " is the " } {XPPEDIT 339 1 "j,i^th" "6$%\"jG)%\"iG%#thG" }{TEXT -1 13 " entry of \+ " }{XPPEDIT 338 1 "A" "I\"AG6\"" }{TEXT 340 1 ":" }{TEXT -1 6 " \+ " }{XPPEDIT 365 1 "A^t*[i,j]=A[j,i" "/*&)%\"AG%\"tG\"\"\"7$%\"iG%\"jG F'&F%6$F*F)" }{TEXT -1 1 " " }{TEXT 366 1 "." }{TEXT -1 14 " The map " }{XPPEDIT 324 1 "A->A^t" ":6#%\"AG7\"6$%)operatorG%&arrowG6\")F$ %\"tGF)F)" }{TEXT -1 18 " is called the " }{TEXT 341 21 "transpose \+ mapping " }{TEXT -1 7 "from " }{XPPEDIT 355 1 "M^(m,n)" ")%\"MG6$ %\"mG%\"nG" }{TEXT -1 8 " to " }{XPPEDIT 356 1 "M^(n,m)" ")%\"MG6$ %\"nG%\"mG" }{TEXT 357 1 "." }{TEXT -1 5 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 353 94 "______________________________________________________________ ________________________________" }}{PARA 0 "" 0 "" {TEXT 342 6 "Maple :" }{TEXT -1 52 " How is the transpose of a matrix obtained using \+ " }{TEXT 347 5 "Maple" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 343 9 "Solution:" }{TEXT -1 23 " There is \+ a command, " }{TEXT 345 13 " transpose(A)" }{TEXT -1 6 ", in " } {TEXT 344 5 "Maple" }{TEXT -1 31 " that does this. It is in the " } {TEXT 346 6 "linalg" }{TEXT -1 41 " package. One can therefore load the " }{TEXT 354 6 "linalg" }{TEXT -1 41 " package and subsequently make the call " }{TEXT 348 15 " transpose(A)" }{TEXT -1 47 " in s hort form or one can use the long form " }{TEXT 349 20 "linalg[transp ose](A)" }{TEXT -1 29 " without first loading the " }{TEXT 350 6 "li nalg" }{TEXT -1 9 " package" }{TEXT 351 1 "." }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, ne w definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new def inition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A := m atrix( [[1,2,3,4], [5,6,7,8]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AG-%'MATRIXG6#7$7&\"\"\"\"\"#\"\"$\"\"%7&\"\"&\"\"'\"\"(\"\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "transpose(A);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7$\"\"\"\"\"&7$\"\"#\"\"'7$\"\"$\" \"(7$\"\"%\"\")" }}}{PARA 0 "" 0 "" {TEXT 352 94 "____________________ ______________________________________________________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "The following theorem on transposes is ea sily verified:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 358 8 "Theorem:" }{TEXT -1 32 " \+ \n1) The transpose mapping " }{XPPEDIT 367 1 "A ->A^t" ":6#%\"AG7 \"6$%)operatorG%&arrowG6\")F$%\"tGF)F)" }{TEXT -1 10 " from " } {XPPEDIT 368 1 "M^(m,n)" ")%\"MG6$%\"mG%\"nG" }{TEXT -1 8 " to " } {XPPEDIT 369 1 "M^(n,m)" ")%\"MG6$%\"nG%\"mG" }{TEXT -1 15 " is l inear" }{TEXT 370 1 "." }{TEXT -1 71 " \n2) Every matrix equals the tr anspose of its transpose:\n\n " }{XPPEDIT 359 1 "`(`*A^t*` )`^t = A" "/*(%\"(G\"\"\")%\"AG%\"tGF%)%\")GF(F%F'" }{TEXT -1 1 " " } {TEXT 360 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "\n\n3) If the matrix pr oduct " }{XPPEDIT 361 1 "A*B" "*&%\"AG\"\"\"%\"BGF$" }{TEXT -1 42 " \+ is defined, then the matrix product " }{XPPEDIT 362 1 "B^t*A^t" " *&)%\"BG%\"tG\"\"\")%\"AGF%F&" }{TEXT -1 34 " is defined and \n\n \n . " }{XPPEDIT 363 1 "(A*B)^t=B^t*A^t" "/)*&%\"AG\"\"\"%\"BG F&%\"tG*&)F'F(F&)F%F(F&" }{TEXT -1 1 " " }{TEXT 364 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "4) If " } {XPPEDIT 371 1 "A " "I\"AG6\"" }{TEXT -1 23 " is any matrix in " }{XPPEDIT 374 1 "M^(m,n)" ")%\"MG6$%\"mG%\"nG" }{TEXT -1 13 " (and s o " }{XPPEDIT 375 1 "A " "I\"AG6\"" }{TEXT -1 99 " is not necessa rily square) then the matrix products\n \n \+ " }{XPPEDIT 372 1 "A*A^t" "*&%\"AG\"\"\")F#%\"tGF$" } {TEXT -1 12 " and " }{XPPEDIT 373 1 "A^t*A" "*&)%\"AG%\"tG\"\" \"F$F&" }{TEXT -1 33 " \n\nare defined and result in " }{TEXT 385 6 "square" }{TEXT -1 10 " matrices" }{TEXT 386 1 ":" }{TEXT -1 17 " \+ the matrix " }{XPPEDIT 376 1 "A*A^t" "*&%\"AG\"\"\")F#%\"tGF$" } {TEXT -1 7 " is " }{XPPEDIT 379 1 "m*` x `*m " "*(%\"mG\"\"\"%$~x~G F$F#F$" }{TEXT -1 19 " and the matrix " }{XPPEDIT 377 1 "A^t*A" "*& )%\"AG%\"tG\"\"\"F$F&" }{TEXT -1 7 " is " }{XPPEDIT 380 1 "n*` x `* n " "*(%\"nG\"\"\"%$~x~GF$F#F$" }{TEXT 378 1 "." }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 384 94 "________________________________________________________ ______________________________________" }}{PARA 0 "" 0 "" {TEXT 381 6 "Maple:" }{TEXT -1 8 " Use " }{TEXT 383 5 "Maple" }{TEXT -1 18 " \+ to calculate " }{XPPEDIT 387 1 "S=A*A^t" "/%\"SG*&%\"AG\"\"\")F%%\"t GF&" }{TEXT -1 9 " and " }{XPPEDIT 389 1 "T=A^t*A" "/%\"TG*&)%\"AG %\"tG\"\"\"F&F(" }{TEXT -1 12 " when " }{XPPEDIT 388 1 "A = MATR IX([[1, 2, 3, 4], [5, 6, 7, 8]])" "/%\"AG-%'MATRIXG6#7$7&\"\"\"\"\"#\" \"$\"\"%7&\"\"&\"\"'\"\"(\"\")" }{TEXT 390 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Do " }{XPPEDIT 391 1 "S" "I\"SG6\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 392 1 "T" "I\"TG6\"" }{TEXT -1 54 " have any special prop erties other than being square?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 382 9 "Solution: " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "A := matrix([[1, 2, 3, 4], [5, 6, 7, 8]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7$7&\"\"\"\"\"#\"\"$ \"\"%7&\"\"&\"\"'\"\"(\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "S := evalm( A &* transpose(A) );" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"SG-%'MATRIXG6#7$7$\"#I\"#q7$F+\"$u\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "T := evalm( transpose(A) &* A);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"TG-%'MATRIXG6#7&7&\"#E\"#K\"#Q\"#W7&F+\"#S\"#[\"# c7&F,F0\"#e\"#o7&F-F1F4\"#!)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Observe t hat both " }{XPPEDIT 394 1 "S" "I\"SG6\"" }{TEXT -1 7 " and " } {XPPEDIT 395 1 "T" "I\"TG6\"" }{TEXT -1 268 " are symmetric about th e principal diagonal: for every entry above the diagonal there is an \+ equal value in the entry that corresponds to a reflection through the \+ principal diagonal. Another way to say the same thing is that both mat rices equal their own transposes:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm( S - transpose(S) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'M ATRIXG6#7$7$\"\"!F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ev alm( T - transpose(T) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG 6#7&7&\"\"!F(F(F(F'F'F'" }}}{PARA 0 "" 0 "" {TEXT 393 93 "____________ ______________________________________________________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "2. Symmetric Matrices and Anti-Symmetric Matrices" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 396 11 "Definition:" }{TEXT -1 14 " A matrix " } {XPPEDIT 397 1 "A" "I\"AG6\"" }{TEXT -1 18 " is said to be " } {TEXT 398 9 "symmetric" }{TEXT -1 7 " or " }{TEXT 411 14 "self-tran spose" }{TEXT -1 10 " if " }{XPPEDIT 399 1 "A=A^t" "/%\"AG)F#%\" tG" }{TEXT 400 1 "." }{TEXT -1 50 " (Clearly a symmetric matrix mus t be square.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 401 8 "Insight:" }{TEXT -1 17 " If a matrix " }{XPPEDIT 402 1 "A" "I\"AG6\"" }{TEXT -1 24 " is symmetric then " }{XPPEDIT 403 1 "A[i,j]=A^t*[i,j]" "/&%\"AG6$%\"iG%\"jG*&)F$%\"tG\"\"\"7$F&F'F+ " }{TEXT -1 10 ". But " }{XPPEDIT 404 1 "A^t*[i,j]=A[j,i]" "/*&)% \"AG%\"tG\"\"\"7$%\"iG%\"jGF'&F%6$F*F)" }{TEXT -1 17 " . Therefore \+ " }{XPPEDIT 405 1 "A[i,j]=A[j,i]" "/&%\"AG6$%\"iG%\"jG&F$6$F'F&" } {TEXT -1 22 " . which states that " }{XPPEDIT 406 1 "A" "I\"AG6\"" } {TEXT -1 5 " is " }{TEXT 407 9 "symmetric" }{TEXT -1 65 " (in the usu al nontechnical sense) about its principal diagonal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 408 8 "Theorem:" }{TEXT -1 16 " \n1) If " }{XPPEDIT 413 1 "A" "I\"AG6\"" }{TEXT -1 6 " is " }{TEXT 415 3 "any" }{TEXT -1 18 " matrix then " }{XPPEDIT 414 1 "A*A^t" "*&%\"AG\"\"\")F#%\"tGF$ " }{TEXT -1 13 " and " }{XPPEDIT 416 1 "A^t*A" "*&)%\"AG%\"tG \"\"\"F$F&" }{TEXT -1 40 " are symmetric matrices.\n\n2) If " }{XPPEDIT 409 1 "A" "I\"AG6\"" }{TEXT -1 11 " is any " }{TEXT 412 6 "square" }{TEXT -1 18 " matrix then " }{XPPEDIT 410 1 "(A+A^t)/ 2" "*&,&%\"AG\"\"\")F$%\"tGF%F%\"\"#!\"\"" }{TEXT -1 18 " is symme tric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 459 94 "______________________________________________ ________________________________________________" }}{PARA 0 "" 0 "" {TEXT 455 6 "Maple:" }{TEXT -1 12 " How is " }{XPPEDIT 460 1 "(A+A ^t)/2" "*&,&%\"AG\"\"\")F$%\"tGF%F%\"\"#!\"\"" }{TEXT -1 17 " obtain ed Use " }{TEXT 457 5 "Maple" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 456 9 "Solution:" }{TEXT -1 29 " The following homebrewed " }{TEXT 462 5 "Maple" }{TEXT -1 21 " fu nction returns " }{XPPEDIT 461 1 "(A+A^t)/2" "*&,&%\"AG\"\"\")F$%\"t GF%F%\"\"#!\"\"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 361 "symmPart :=\n \nproc()\nloc al A, symmetricPart:\n \nif nargs <> 1 then\n ERROR(`symm Part expects one argument.`)\n\nelif not type(args[1],'matrix'(square) ) then\n ERROR(`symmPart expects its argument to be a square matrix. `);\n\nelse\n A := args[1];\nfi:\n \nsymmetricPart := linalg[scalar mul](A+linalg[transpose](A),1/2):\nRETURN(evalm(symmetricPart));\nend; \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)symmPartG:6\"6$%\"AG%.symmetr icPartGF&F&C%@'09#\"\"\"-%&ERRORG6#%?symmPart~expects~one~argument.G4- %%typeG6$&9\"6#F.-.%'matrixG6#%'squareG-F06#%UsymmPart~expects~its~arg ument~to~be~a~square~matrix.G>8$F7>8%-&%'linalgG6#%*scalarmulG6$,&FCF. -&FH6#%*transposeG6#FCF.#F.\"\"#-%'RETURNG6#-%&evalmG6#FEF&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := matrix( [[1,2,3], [4,5, 6], [7,8,9]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7% 7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "symmPart(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"$\"\"&7%F)F*\"\"(7%F*F,\"\"* " }}}{PARA 0 "" 0 "" {TEXT 458 94 "___________________________________ ___________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 417 11 "Definition:" }{TEXT -1 14 " A matrix " }{XPPEDIT 418 1 "A" "I \"AG6\"" }{TEXT -1 18 " is said to be " }{TEXT 419 14 "anti-symmetr ic" }{TEXT -1 7 " or " }{TEXT 438 14 "skew-symmetric" }{TEXT -1 10 " if " }{XPPEDIT 420 1 "A^t=-A" "/)%\"AG%\"tG,$F$!\"\"" }{TEXT 421 1 "." }{TEXT -1 55 " (Clearly a skew-symmetric matrix must be s quare.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 422 8 "Insight:" }{TEXT -1 17 " If a matrix " }{XPPEDIT 423 1 "A" "I\"AG6\"" }{TEXT -1 29 " is skew-symmetric then " }{XPPEDIT 424 1 "A[i,j]=-A^t*[i,j]" "/&%\"AG6$%\"iG%\"jG,$*&)F$%\"tG\"\"\"7$F&F'F,! \"\"" }{TEXT -1 10 ". But " }{XPPEDIT 425 1 "A^t*[i,j]=A[j,i]" "/* &)%\"AG%\"tG\"\"\"7$%\"iG%\"jGF'&F%6$F*F)" }{TEXT -1 17 " . Therefore " }{XPPEDIT 426 1 "A[i,j]=-A[j,i]" "/&%\"AG6$%\"iG%\"jG,$&F$6$F'F& !\"\"" }{TEXT -1 132 " . The entries below the principal diagonal are opposite in sign to those in corresponding positions above the princi pal diagonal.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 427 8 "Theorem:" }{TEXT -1 13 " \+ \n If " }{XPPEDIT 428 1 "A" "I\"AG6\"" }{TEXT -1 11 " is any \+ " }{TEXT 439 6 "square" }{TEXT -1 18 " matrix then " }{XPPEDIT 429 1 "(A-A^t)/2" "*&,&%\"AG\"\"\")F$%\"tG!\"\"F%\"\"#F(" }{TEXT -1 23 " is skew-symmetric." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 434 94 "________________________________________________________ ______________________________________" }}{PARA 0 "" 0 "" {TEXT 430 6 "Maple:" }{TEXT -1 12 " How is " }{XPPEDIT 435 1 "(A-A^t)/2" "*&,& %\"AG\"\"\")F$%\"tG!\"\"F%\"\"#F(" }{TEXT -1 19 " obtained using " }{TEXT 432 5 "Maple" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 431 9 "Solution:" }{TEXT -1 29 " The followi ng homebrewed " }{TEXT 437 5 "Maple" }{TEXT -1 21 " function returns " }{XPPEDIT 436 1 "(A+A^t)/2" "*&,&%\"AG\"\"\")F$%\"tGF%F%\"\"#!\" \"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 373 "skewSymmPart :=\n \nproc()\nlocal A, sym metricPart:\n \nif nargs <> 1 then\n ERROR(`skewSymmPart \+ expects one argument.`)\n\nelif not type(args[1],'matrix'(square)) the n\n ERROR(`skewSymmPart expects its argument to be a square matrix.` );\n\nelse\n A := args[1];\nfi:\n \nsymmetricPart := linalg[scalarm ul](A-linalg[transpose](A),1/2):\nRETURN(evalm(symmetricPart));\nend; \n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-skewSymmPartG:6\"6$%\"AG%.sym metricPartGF&F&C%@'09#\"\"\"-%&ERRORG6#%CskewSymmPart~expects~one~argu ment.G4-%%typeG6$&9\"6#F.-.%'matrixG6#%'squareG-F06#%YskewSymmPart~exp ects~its~argument~to~be~a~square~matrix.G>8$F7>8%-&%'linalgG6#%*scalar mulG6$,&FCF.-&FH6#%*transposeG6#FC!\"\"#F.\"\"#-%'RETURNG6#-%&evalmG6# FEF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A := matrix( [[1, 2,3], [4,5,6], [7,8,9]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%' MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "skewSymmPart(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"!!\"\"!\"#7%\"\"\"F(F)7% \"\"#F,F(" }}}{PARA 0 "" 0 "" {TEXT 433 94 "__________________________ ____________________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 440 8 "Theorem:" }{TEXT -1 13 " \n If " } {XPPEDIT 441 1 "A" "I\"AG6\"" }{TEXT -1 11 " is any " }{TEXT 443 6 "square" }{TEXT -1 16 " matrix then " }{XPPEDIT 444 1 "A" "I\"AG6\" " }{TEXT -1 64 " is the sum of a symmetric matrix and an anti-symmet ric matrix" }{TEXT 446 1 ":" }{TEXT -1 45 " \n\n \+ " }{XPPEDIT 442 1 "A=(A+A^t)/2+(A-A^t)/2" "/%\"AG, &*&,&F#\"\"\")F#%\"tGF'F'\"\"#!\"\"F'*&,&F#F')F#F)F+F'F*F+F'" }{TEXT -1 1 " " }{TEXT 445 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 451 94 "______________________ ______________________________________________________________________ __" }}{PARA 0 "" 0 "" {TEXT 447 6 "Maple:" }{TEXT -1 9 " Let " } {XPPEDIT 452 1 "A" "I\"AG6\"" }{TEXT -1 31 " be any random matrix. \+ Use " }{TEXT 449 5 "Maple" }{TEXT -1 31 " to verify the last theorem . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 448 9 "So lution:" }{TEXT -1 29 " The following homebrewed " }{TEXT 454 5 "Ma ple" }{TEXT -1 21 " function returns " }{XPPEDIT 453 1 "(A+A^t)/2" "*&,&%\"AG\"\"\")F$%\"tGF%F%\"\"#!\"\"" }{TEXT -1 3 " . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n := \+ rand(3..5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"nG:6\"6#%\"tGF&F&C% >%&_seedG-%%iremG6$,$F+\"-\"3p'>uU\"-*)**********>8$F+,&-F-6$F3\"\"$\" \"\"F7F8F&6#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "m := n(); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "A := randmatrix(m,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7'7'\"#%*\"#$)!#')\"#B!#%)7'\"#>!#]\" #))!#`\"#&)7'\"#\\\"#y\"#<\"#s!#**7'!#&)F,\"#I\"#!)F97'\"#m!#H!#\"*F3! #>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sP := symmPart(A);\ns SP :=skewSymmPart(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sPG-%'MATR IXG6#7'7'\"#%*\"#^#!#P\"\"#!#J!\"*7'F+!#]\"#$)#!$R\"F.\"#G7'F,F3\"#F.7'F0F6F9F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sSPG-%'MATRIXG6#7'7'\"\"!\"#K#!$N\"\"\"#\"#a!#v7'!#KF*\"\"&#\"#L F.\"#d7'#\"$N\"F.!\"&F*\"#@!\"%7'!#a#!#LF.!#@F*#\"$D\"F.7'\"#v!#d\"\"% #!$D\"F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm( A - (s P + sSP) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"!F( F(F(F(F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT 450 94 "_____________________ ______________________________________________________________________ ___" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 " 3. Bilinear Maps" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 11 "Definition:" }{TEXT -1 9 " Let " }{TEXT 263 1 "U" } {TEXT -1 3 ", " }{TEXT 264 1 "V" }{TEXT -1 6 ", and " }{TEXT 265 1 "W " }{TEXT -1 18 " be vector spaces" }{TEXT 552 1 "." }{TEXT -1 14 " \+ A function " }{TEXT 266 3 " B " }{TEXT -1 13 " with domain " }{TEXT 267 7 " U x V " }{TEXT -1 13 " and range " }{TEXT 268 1 "W" }{TEXT -1 18 " is said to be " }{TEXT 269 10 "\n\nbilinear" }{TEXT -1 7 " \+ if\n" }}{PARA 0 "" 0 "" {TEXT -1 6 "1) " }{XPPEDIT 274 1 "u->B(u ,v)" ":6#%\"uG7\"6$%)operatorG%&arrowG6\"-%\"BG6$F$%\"vGF)F)" }{TEXT 275 1 " " }{TEXT -1 23 " is a linear map from " }{TEXT 553 3 " U " } {TEXT -1 6 " to " }{TEXT 554 1 "W" }{TEXT -1 13 " for each " } {TEXT 272 4 " v " }{TEXT -1 3 "in " }{TEXT 273 2 " V" }{TEXT -1 6 " \+ and" }}{PARA 0 "" 0 "" {TEXT -1 6 "2) " }{XPPEDIT 276 1 "v->B(u,v) " ":6#%\"vG7\"6$%)operatorG%&arrowG6\"-%\"BG6$%\"uGF$F)F)" }{TEXT 277 1 " " }{TEXT -1 22 " is a linear map from " }{TEXT 555 5 " V " } {TEXT -1 6 " to " }{TEXT 556 1 "W" }{TEXT -1 12 " for each " } {TEXT 271 3 " u" }{TEXT -1 6 " in " }{TEXT 270 1 "U" }{TEXT 278 1 " ." }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 279 8 "Example:" }{TEXT -1 8 " Let " }{XPPEDIT 280 1 "U=M^( m,n)" "/%\"UG)%\"MG6$%\"mG%\"nG" }{TEXT -1 19 " (respectively " } {XPPEDIT 282 1 "V=M^(n,p)" "/%\"VG)%\"MG6$%\"nG%\"pG" }{TEXT -1 21 ", \+ respectively " }{XPPEDIT 285 1 "W=M^(m,p)" "/%\"WG)%\"MG6$%\"mG% \"pG" }{TEXT -1 21 ") be the space of " }{XPPEDIT 281 1 "m*` x `*n \+ " "*(%\"mG\"\"\"%$~x~GF$%\"nGF$" }{TEXT -1 21 " (respectively " }{XPPEDIT 283 1 "n*` x `*p" "*(%\"nG\"\"\"%$~x~GF$%\"pGF$" }{TEXT -1 19 ", respectively " }{XPPEDIT 286 1 "m*` x `*p" "*(%\"mG\"\"\"%$~ x~GF$%\"pGF$" }{TEXT -1 23 ") matrices. Then " }{XPPEDIT 284 1 " B(S,T)=S*T" "/-%\"BG6$%\"SG%\"TG*&F&\"\"\"F'F)" }{TEXT -1 45 " (prod uct of matrices) is bilinear. If " }{XPPEDIT 287 1 "T" "I\"TG6\"" }{TEXT -1 25 " is fixed then the map " }{XPPEDIT 288 1 "S ->B(S,T)" ":6#%\"SG7\"6$%)operatorG%&arrowG6\"-%\"BG6$F$%\"TGF)F)" }{TEXT -1 40 " satisfies \n\n " }{XPPEDIT 289 1 "B(a[1]*S [1]+a[2]*S[2],T)=(a[1]*S[1]+a[2]*S[2])*T" "/-%\"BG6$,&*&&%\"aG6#\"\"\" F+&%\"SG6#F+F+F+*&&F)6#\"\"#F+&F-6#F2F+F+%\"TG*&,&*&&F)6#F+F+&F-6#F+F+ F+*&&F)6#F2F+&F-6#F2F+F+F+F5F+" }{TEXT -1 43 " \nwhich equals \n \+ " }{XPPEDIT 290 1 "a[1]*S[1]*T+a[2]*S[2]*T" ",&*( &%\"aG6#\"\"\"F'&%\"SG6#F'F'%\"TGF'F'*(&F%6#\"\"#F'&F)6#F/F'F+F'F'" } {TEXT -1 3 " \n" }}{PARA 0 "" 0 "" {TEXT -1 51 "after distribution. T his last expression equals " }{XPPEDIT 291 1 "a[1]*B(S[1],T)+a[2]*B (S[2],T)" ",&*&&%\"aG6#\"\"\"F'-%\"BG6$&%\"SG6#F'%\"TGF'F'*&&F%6#\"\"# F'-F)6$&F,6#F2F.F'F'" }{TEXT -1 35 " which proves that\n\n \+ " }{XPPEDIT 292 1 "B(a[1]*S[1]+a[2]*S[2],T)=a[1]*B(S[1],T)+a[2]*B(S [2],T)" "/-%\"BG6$,&*&&%\"aG6#\"\"\"F+&%\"SG6#F+F+F+*&&F)6#\"\"#F+&F-6 #F2F+F+%\"TG,&*&&F)6#F+F+-F$6$&F-6#F+F5F+F+*&&F)6#F2F+-F$6$&F-6#F2F5F+ F+" }{TEXT -1 1 " " }{TEXT 293 1 "." }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 75 "\nLinearity in the second variable is established in pr ecisely the same way." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 8 "Example:" }{TEXT -1 8 " Let " }{XPPEDIT 295 1 "U,V,W" "6%%\"UG%\"VG%\"WG" }{TEXT -1 24 " all equal the space " }{XPPEDIT 297 1 "M^(n,n)" ")%\"MG6$%\"nGF%" }{TEXT -1 13 " of square " } {XPPEDIT 296 1 "n*` x `*n " "*(%\"nG\"\"\"%$~x~GF$F#F$" }{TEXT -1 22 " matrices. Let " }{XPPEDIT 298 1 "B(S,T)=S+T" "/-%\"BG6$%\"SG% \"TG,&F&\"\"\"F'F)" }{TEXT -1 29 " (sum of matrices). Then " } {XPPEDIT 301 1 "B" "I\"BG6\"" }{TEXT -1 6 " is " }{TEXT 302 3 "not" }{TEXT -1 18 " bilinear. If " }{XPPEDIT 299 1 "T" "I\"TG6\"" } {TEXT -1 36 " is a fixed nonzero matrix then " }{XPPEDIT 300 1 "B( S[1]+S[2],T)=S[1]+S[2]+2*T" "/-%\"BG6$,&&%\"SG6#\"\"\"F*&F(6#\"\"#F*% \"TG,(&F(6#F*F*&F(6#F-F**&F-F*F.F*F*" }{TEXT -1 13 " whereas " } {XPPEDIT 303 1 "B(S[1],T)+B(S[2],T)=S[1]+S[2]+T" "/,&-%\"BG6$&%\"SG6# \"\"\"%\"TGF*-F%6$&F(6#\"\"#F+F*,(&F(6#F*F*&F(6#F0F*F+F*" }{TEXT -1 1 " " }{TEXT 304 1 "." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 305 26 "Definition (Extra Jargon):" }{TEXT -1 30 " A bilinear map with domain " }{XPPEDIT 310 1 "V*` x `*V" "*(%\" VG\"\"\"%$~x~GF$F#F$" }{TEXT -1 42 " and with values in the ground fi eld of " }{XPPEDIT 311 1 "V" "I\"VG6\"" }{TEXT -1 19 " is said to be \+ a \n\n" }{TEXT 307 35 " bilinear functional" }{TEXT -1 15 " or " }{TEXT 306 23 " bilinear form" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "on " }{XPPEDIT 316 1 "V" "I\"VG6\" " }{TEXT -1 1 " " }{TEXT 317 1 "." }{TEXT -1 3 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 10 "Example 1:" }{TEXT -1 8 " Let " }{XPPEDIT 309 1 "V=R^ n" "/%\"VG)%\"RG%\"nG" }{TEXT -1 68 ", the space of real column vect ors. The dot product\n\n " }{XPPEDIT 312 1 "B(` `*MATRIX([ [x[1]],[x[2]],[`.`],[`.`],[x[n]]])*` `, ` `*MATRIX([[y[1]],[y[2]],[`.` ],[`.`],[y[n]]]) *` `) =x[1]*y[1]+x[2]*y[2]+`...`+x[n]*y[n]" "/-%\"BG6 $*(%\"~G\"\"\"-%'MATRIXG6#7'7#&%\"xG6#F(7#&F/6#\"\"#7#%\".G7#F67#&F/6# %\"nGF(F'F(*(F'F(-F*6#7'7#&%\"yG6#F(7#&FB6#F47#F67#F67#&FB6#F;F(F'F(,* *&&F/6#F(F(&FB6#F(F(F(*&&F/6#F4F(&FB6#F4F(F(%$...GF(*&&F/6#F;F(&FB6#F; F(F(" }{TEXT -1 118 " \n\nis a bilinear form. It is conveni ent to write this bilinear form as the matrix product:\n\n\n \+ " }{XPPEDIT 463 1 "B(` `*MATRIX([[x[1]],[x[2]],[`.`],[`.`],[x[n]]] )*` `, ` `*MATRIX([[y[1]],[y[2]],[`.`],[`.`],[y[n]]]) *` `) =[x[1],x[2 ],`...`,x[n]]*MATRIX( [[y[1]],[y[2]],[`.`],[`.`],[`.`],[y[n]]])" "/-% \"BG6$*(%\"~G\"\"\"-%'MATRIXG6#7'7#&%\"xG6#F(7#&F/6#\"\"#7#%\".G7#F67# &F/6#%\"nGF(F'F(*(F'F(-F*6#7'7#&%\"yG6#F(7#&FB6#F47#F67#F67#&FB6#F;F(F 'F(*&7&&F/6#F(&F/6#F4%$...G&F/6#F;F(-F*6#7(7#&FB6#F(7#&FB6#F47#F67#F67 #F67#&FB6#F;F(" }{TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "In other words" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " \+ " }{XPPEDIT 464 1 "B(x,y) = x^t*``*y" "/-%\"BG6$%\"xG%\"yG*()F& %\"tG\"\"\"%!GF+F'F+" }{TEXT -1 1 " " }{TEXT 466 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "It is also useful to n ote that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " " }{XPPEDIT 465 1 "B(x,y) = x^t*`` *I[nxn]*``*y" "/-%\"BG6$%\"xG%\"yG*,)F&%\"tG\"\"\"%!GF+&%\"IG6#%$nxnGF +F,F+F'F+" }{TEXT -1 4 " ." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 8 "where " }{XPPEDIT 467 1 "I[nxn]" "&%\"IG 6#%$nxnG" }{TEXT -1 12 " is the " }{XPPEDIT 468 1 "n*``*x*``*n" "* ,%\"nG\"\"\"%!GF$%\"xGF$F%F$F#F$" }{TEXT -1 19 " identity matrix." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 560 93 "______________ ______________________________________________________________________ _________" }}{PARA 0 "" 0 "" {TEXT 557 6 "Maple:" }{TEXT -1 42 " Ho w is the dot product calculated in " }{TEXT 559 5 "Maple" }{TEXT -1 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 558 9 " Solution:" }{TEXT -1 23 " There is a builtin " }{TEXT 561 5 "Maple " }{TEXT -1 13 " function, " }{TEXT 563 7 "dotprod" }{TEXT -1 24 ", \+ for the dot product. " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}{PARA 7 " " 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 " " {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x := vector([-1,2,3,4,5]); y := vector([a,b, c,d,e]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'VECTORG6#7'!\"\" \"\"#\"\"$\"\"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'VECTO RG6#7'%\"aG%\"bG%\"cG%\"dG%\"eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%\"aG!\"\"% \"bG\"\"#%\"cG\"\"$%\"dG\"\"%%\"eG\"\"&" }}}{PARA 0 "" 0 "" {TEXT 562 93 "__________________________________________________________________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 10 "Example 2:" }{TEXT -1 56 " We can easily generalize the preceding example. Let " } {XPPEDIT 318 1 "V=R^n" "/%\"VG)%\"RG%\"nG" }{TEXT -1 53 ", the space of real column vectors. Suppose that " }{XPPEDIT 315 1 "d[1],d[2],` ...`,d[n]" "6&&%\"dG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 41 " are scalars. Then the map \n\n " }{XPPEDIT 314 1 "B(` `*MA TRIX([[x[1]],[x[2]],[`.`],[`.`],[x[n]]])*` `, ` `*MATRIX([[y[1]],[y[2] ],[`.`],[`.`],[y[n]]]) *` `) =d[1]*x[1]*y[1]+d[2]*x[2]*y[2]+`...`+d[n] *x[n]*y[n]" "/-%\"BG6$*(%\"~G\"\"\"-%'MATRIXG6#7'7#&%\"xG6#F(7#&F/6#\" \"#7#%\".G7#F67#&F/6#%\"nGF(F'F(*(F'F(-F*6#7'7#&%\"yG6#F(7#&FB6#F47#F6 7#F67#&FB6#F;F(F'F(,**(&%\"dG6#F(F(&F/6#F(F(&FB6#F(F(F(*(&FO6#F4F(&F/6 #F4F(&FB6#F4F(F(%$...GF(*(&FO6#F;F(&F/6#F;F(&FB6#F;F(F(" }{TEXT -1 33 " \n\nis a bilinear form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "Notice that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " " }{XPPEDIT 469 1 "B(x,y) = x^t*``*A*``*y" "/-%\"BG6$%\"xG%\"yG*,)F&%\"t G\"\"\"%!GF+%\"AGF+F,F+F'F+" }{TEXT -1 4 " ." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "where " }{XPPEDIT 470 1 "A" "I\"AG6\"" }{TEXT -1 12 " is the " }{XPPEDIT 471 1 "n*``*x*` `*n" "*,%\"nG\"\"\"%!GF$%\"xGF$F%F$F#F$" }{TEXT -1 74 " diagonal mat rix:\n\n " } {XPPEDIT 472 1 "MATRIX([ [d[1],0,0,0,0,0],[0,d[2],0,0,0,0],[0,0,`.`,0, 0,0],[0,0,0,`.`,0,0],[0,0,0,0,`.`,0],[0,0,0,0,0,d[n]] ])" "-%'MA TRIXG6#7(7(&%\"dG6#\"\"\"\"\"!F+F+F+F+7(F+&F(6#\"\"#F+F+F+F+7(F+F+%\". GF+F+F+7(F+F+F+F1F+F+7(F+F+F+F+F1F+7(F+F+F+F+F+&F(6#%\"nG" }{TEXT -1 4 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 473 10 "Example \+ 3:" }{TEXT -1 56 " We can easily generalize the preceding example. Le t " }{XPPEDIT 476 1 "V=R^n" "/%\"VG)%\"RG%\"nG" }{TEXT -1 53 ", th e space of real column vectors. Suppose that " }{XPPEDIT 475 1 "A" " I\"AG6\"" }{TEXT -1 11 " is any " }{XPPEDIT 477 1 "n*` x `*n" "*(% \"nG\"\"\"%$~x~GF$F#F$" }{TEXT -1 39 " matrix. Then the map \n\n \+ " }{XPPEDIT 474 1 "B(x,y)=x^t*``*A*``*y" "/-%\"BG6$%\"xG%\"yG *,)F&%\"tG\"\"\"%!GF+%\"AGF+F,F+F'F+" }{TEXT -1 33 " \n\nis a bilinear form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 482 93 "__________________________________________________________________ ___________________________" }}{PARA 0 "" 0 "" {TEXT 478 6 "Maple:" } {TEXT -1 9 " Use " }{TEXT 480 5 "Maple" }{TEXT -1 80 " to explici tly write out the bilinear form associated with the square matrix " }{XPPEDIT 484 1 "MATRIX([[1, 0, -2], [3, 2, 1], [0, 0, 4]])" "-%'MATRI XG6#7%7%\"\"\"\"\"!,$\"\"#!\"\"7%\"\"$F*F'7%F(F(\"\"%" }{TEXT -1 6 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 479 9 "S olution:" }{TEXT -1 23 " There is a builtin " }{TEXT 564 5 "Maple" }{TEXT -1 12 " function, " }{TEXT 565 9 "innerprod" }{TEXT -1 17 ", f or doing this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: wit h(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trac e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A := matrix([[1, 0, -2 ], [3, 2, 1], [0, 0, 4]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-% 'MATRIXG6#7%7%\"\"\"\"\"!!\"#7%\"\"$\"\"#F*7%F+F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x := vector(3); y := vector(3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%&arrayG6$;\"\"\"\"\"$7\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%&arrayG6$;\"\"\"\"\"$7\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "innerprod(x,A,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"yG6#\"\"\"F(&%\"xGF'F(F(*&F%F(&F*6# \"\"#F(\"\"$*&F,F(&F&F-F(F.*&&F&6#F/F(F)F(!\"#*&F3F(F,F(F(*&F3F(&F*F4F (\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "The following homebrewed " } {TEXT 483 5 "Maple" }{TEXT -1 69 " function can be used quite general ly - in this context and others. " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2809 "bilinearForm := \n\nproc( )\nlocal A, n, x, y;\n\n############################################## ###########\n# # \n# Case 1: Wrong number of arguments #\n# \+ #\n#################### #####################################\n\nif not member(nargs,\{3,4\}) \+ then\nERROR(`bilinearForm expects three or four arguments.`):\n\n\n### ######################################################\n# \+ #\n# Case 2: Three arg uments #\n# \+ #\n# This case is similar to innerprod(x,A,y) e xcept #\n# the calling sequence is bilinearForm(A,x,y) # \+ \n# #\n######### ################################################\n\nelif nargs = 3 the n\n\nA := args[1]:\nx := args[2]:\ny := args[3]:\n\n if not type(A, \+ 'matrix'(square)) then\n ERROR(`bilinearForm expects its first argum ent to be a square matrix.`)\n fi:\n\nn := linalg[rowdim](A);\n\n \+ if not type(x,\{vector,list\}) or \n not type(y,\{vector,list\}) \+ then\n ERROR(`bilinearForm expects each of its last two arguments to be a vector or a list.`)\n\n else\n x := convert(x,vector):\n y := convert(y,vector):\n fi:\n\n if vectdim(x) <> n or \n vec tdim(y) <> n then\n ERROR(`bilinearForm expects its last two argumen ts\nto have an equal number of entries compatible with the size of the first entry.`):\n\n else\n RETURN( simplify(expand(\n \+ evalm(linalg[transpose](x) &* A &* y))) );\n fi:\n \n\n############ #############################################\n# \+ #\n# Case 3: Four arguments \+ #\n# \+ #\n# This case is bilinearForm(f,g,weight,[a,b]) \+ #\n# or bilinearForm(f,g,weight, a..b) # \n# \+ #\n################## #######################################\n\nelse\n\n if not type(args [1],\{name,procedure,constant\}) or \n not type(args[2],\{name,pr ocedure,constant\}) or \n not type(args[3],\{name,procedure,const ant\}) then\n ERROR(`In a four argument call to bilinearForm the fir st three parameters should be procedures.`)\n\n elif not type(args[4 ],\{list,range\}) then\n ERROR(`In a four argument call to bilinearF orm the last argument should be a range or an ordered pair.`);\n \n \+ elif type(args[4],list) and nops(args[4]) <> 2 then\n ERROR(`In a \+ four argument call to bilinearForm the last argument should be a range or an ordered pair.`); \n\n else\n RETURN( int(args[1](x)*args[ 2](x)*args[3](x), \n x = op(1,args[4])..op(2,args[4]) ));\n fi:\n \n\nfi:\nend;" }}{PARA 12 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-bilinearFormG:6\"6&%\"AG%\"nG%\"xG%\"yGF&F&@'4-%'mem berG6$9#<$\"\"$\"\"%-%&ERRORG6#%NbilinearForm~expects~three~or~four~ar guments.G/F1F3C)>8$&9\"6#\"\"\">8&&F>6#\"\"#>8'&F>6#F3@$4-%%typeG6$F<- .%'matrixG6#%'squareG-F66#%inbilinearForm~expects~its~first~argument~t o~be~a~square~matrix.G>8%-&%'linalgG6#%'rowdimG6#F<@%43-FM6$FB<$%%list G%'vectorG-FM6$FGF^o-F66#%hobilinearForm~expects~each~of~its~last~two~ arguments~to~be~a~vector~or~a~list.GC$>FB-%(convertG6$FBF`o>FG-Fio6$FG F`o@%50-%(vectdimG6#FBFX0-Fbp6#FGFX-F66#%frbilinearForm~expects~its~la st~two~arguments|+to~have~an~equal~number~of~entries~compatible~with~t he~size~of~the~first~entry.G-%'RETURNG6#-%)simplifyG6#-%'expandG6#-%&e valmG6#-%#&*G6$-Fgq6$-&Fen6#%*transposeGFcpF6#F4<$F_o%&rangeG-F66#%jpIn~a~four~argument~call~to~bilinearFor m~the~last~argument~should~be~a~range~or~an~ordered~pair.G3-FM6$FcsF_o 0-%%nopsG6#FcsFEFgs-F[q6#-%$intG6$*(-F=FcpF@-FCFcpF@-FHFcpF@/FB;-%#opG 6$F@Fcs-F]u6$FEFcsF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A := matrix([[1, 0, -2], [3, 2, 1], [0, 0, 4]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"\"\"\"\"!!\"#7%\"\"$\"\"#F*7%F+ F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x := vector(3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "y := vector(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "bilinearForm(A,x,y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,.*&&%\"yG6#\"\"\"F(&%\"xGF'F(F(*&F%F(&F*6#\"\"# F(\"\"$*&F,F(&F&F-F(F.*&&F&6#F/F(F)F(!\"#*&F3F(F,F(F(*&F3F(&F*F4F(\"\" %" }}}{PARA 0 "" 0 "" {TEXT 481 94 "__________________________________ ____________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Now that we have seen some examples of how to construct \+ bilinear forms, it is reasonable to wonder when they arise. \n\n" }} {PARA 0 "" 0 "" {TEXT 566 8 "Example:" }{TEXT -1 58 " Bilinear forms are used to create geometric objects:\n\n\n" }{TEXT 567 11 "Sphere: \+ " }{TEXT -1 1 " " }{XPPEDIT 570 1 "x[1]^2+x[2]^2+x[3]^2 = 1" "/,(*$& %\"xG6#\"\"\"\"\"#F(*$&F&6#F)F)F(*$&F&6#\"\"$F)F(F(" }{TEXT 569 14 " \+ or " }{XPPEDIT 19 1 "x^t*I*x=1" "/*()%\"xG%\"tG\"\"\"%\"IGF'F %F'F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 568 16 "Ellipsoid: " }{TEXT -1 1 " " }{XPPEDIT 572 1 "x[1]^2/a^2+x[2]^2/ b^2+x[3]^2/c^2 = 1" "/,(*&&%\"xG6#\"\"\"\"\"#*$%\"aGF)!\"\"F(*&&F&6#F) F)*$%\"bGF)F,F(*&&F&6#\"\"$F)*$%\"cGF)F,F(F(" }{TEXT 571 10 " or \+ " }{TEXT -1 4 " " }{XPPEDIT 573 1 "x^t*MATRIX([[1/a^2,0,0],[0,1/b^ 2,0],[0,0,1/c^2]])*x=1" "/*()%\"xG%\"tG\"\"\"-%'MATRIXG6#7%7%*&F'F'*$% \"aG\"\"#!\"\"\"\"!F27%F2*&F'F'*$%\"bGF0F1F27%F2F2*&F'F'*$%\"cGF0F1F'F %F'F'" }{TEXT -1 4 "\n\n\n\n" }{TEXT 574 18 "Hyperboloid: " } {TEXT -1 1 " " }{XPPEDIT 576 1 "x[1]^2/a^2+x[2]^2/b^2-x[3]^2/c^2 = 0" "/,(*&&%\"xG6#\"\"\"\"\"#*$%\"aGF)!\"\"F(*&&F&6#F)F)*$%\"bGF)F,F(*&&F& 6#\"\"$F)*$%\"cGF)F,F,\"\"!" }{TEXT 575 10 " or " }{TEXT -1 4 " \+ " }{XPPEDIT 577 1 "x^t*MATRIX([[1/a^2,0,0],[0,1/b^2,0],[0,0,-1/c^2] ])*x=0" "/*()%\"xG%\"tG\"\"\"-%'MATRIXG6#7%7%*&F'F'*$%\"aG\"\"#!\"\"\" \"!F27%F2*&F'F'*$%\"bGF0F1F27%F2F2,$*&F'F'*$%\"cGF0F1F1F'F%F'F2" } {TEXT -1 4 "\n\n\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 581 93 "______________________ ______________________________________________________________________ _" }}{PARA 0 "" 0 "" {TEXT 578 6 "Maple:" }{TEXT -1 9 " Use " } {TEXT 580 5 "Maple" }{TEXT -1 53 " to calculate the multivariable Tay lor series of " }{XPPEDIT 582 1 "sqrt(1+x[1]+2*x[2]+3*x[3])" "-%%sq rtG6#,*\"\"\"F&&%\"xG6#F&F&*&\"\"#F&&F(6#F+F&F&*&\"\"$F&&F(6#F/F&F&" } {TEXT -1 67 " . Show that the degree two terms give rise to a bilin ear form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 579 9 "Solution:" }{TEXT -1 23 " There is a builtin " }{TEXT 583 5 "Maple" }{TEXT -1 12 " function, " }{TEXT 584 7 "mtaylor" }{TEXT -1 96 ", for calculating multivariable Taylor series.\nIt must be loaded \+ from the library by means of a " }{TEXT 586 9 " readlib " }{TEXT -1 8 "call. \n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(mtaylor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6\"6*%\"fG%\"kG%\"vG%\"mG%\"nG%\"sG%\"tG%\" wG6#%aoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~rights~re served.GF$C2>8$&9\"6#\"\"\">8&&F46#\"\"#@&-%%typeG6$F8%$setG>F87#-%#op G6#F84-F>6$F8%%listG>F87#F8@$4-F>6$F8-FI6#<$/%%nameG%*algebraicGFT-%&E RRORG6#%Ginvalid~2nd~argument~(expansion~point)G>8)-%$mapG6$:6#%\"xGF$ F$F$@%-F>6$9$%\"=G-%$rhsG6#F_o\"\"!F$F$F8>F8-Fgn6$:FjnF$F$F$@%F]o-%$lh sGFcoF_oF$F$F8>8'-%%nopsGFE@$0F]p-F_p6#<#FC-FW6#%Hvariables~(2nd~argum ent)~must~be~uniqueG@%/9#F;>8(\"\"'>F\\q&F46#\"\"$@%/Fjp\"\"%>8+&F46#F dq>Ffq7#-%\"$G6$F6F]p@$4-F>6$F8<$-F@6#FT-FIFdr-FW6#%O2nd~argument~(the ~variable(s))~must~be~a~namesG@$34-F>6$F\\q%*nonnegintG0F\\q%)infinity G-FW6#%X3rd~argument~(the~order)~must~be~a~non-negative~integerG@$54-F >6$Ffq-FI6#%'posintG0-F_p6#FfqF]p-FW6#%en4th~argument~(weights)~must~b e~a~list~of~positive~integersG>F2-%%subsG6$7#-%$seqG6$/&F86#8%,&*&F[uF 6)8*&FfqF\\uF6F6&FenF\\uF6/F]u;F6F]pF2>F2-Fgn6&%(collectG-Fdt6$/-%\"OG F5Fdo-%'taylorG6%F2FauF\\qF8.%,distributedG>F2-Fdt6$7#-Fht6$/F[u,&F[uF 6Fcu!\"\"Fdu-Fdt6$/FauF6F2F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "multivariableTaylorSeries :=\nmtaylor( sqrt(1+x[1]+2*x[2]+3*x[ 3]) ,[x[1],x[2],x[3]],2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%:multiv ariableTaylorSeriesG,J\"\"\"F&&%\"xG6#F&#F&\"\"#&F(6#F+F&&F(6#\"\"$#F0 F+*$F'F+#!\"\"\"\")*&F.F&F'F&#!\"$\"\"%*&F,F&F'F&#F4F+*$F,F+F;*&F.F&F, F&#F8F+*$F.F+#!\"*F5*$F'F0#F&\"#;*&F.F&F'F+#\"\"*FD*&F,F&F'F+#F0F5*&F' F&F,F+#F0F9*(F.F&F,F&F'F&#FGF9*&F.F+F'F&#\"#FFD*$F,F0F**&F.F&F,F+FM*&F .F+F,F&#FPF5*$F.F0FO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A1 \+ := vector([1/2,1,3/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'VE CTORG6#7%#\"\"\"\"\"#F*#\"\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "A2 := matrix([[-1/8,-1/4,-3/8],[-1/4,-1/2,0],[-3/8,0,-9/8]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'MATRIXG6#7%7%#!\"\"\"\")# F+\"\"%#!\"$F,7%F-#F+\"\"#\"\"!7%F/F4#!\"*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "multivariableTaylorSeries-(1 + dotprod(A1,x) + i nnerprod(x,A2,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,8*&&%\"xG6#\"\" $\"\"#&F&6#\"\"\"F,#\"#F\"#;*$F%F(F-*&F%F,&F&6#F)F)#\"\"*\"\"%*&F%F)F2 F,#F.\"\")*$F2F(#F,F)*&F%F,F2F,#!\"$F)*$F*F(#F,F/*&F2F,F*F)#F(F9*&F%F, F*F)#F5F/*(F%F,F2F,F*F,F4*&F*F,F2F)#F(F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Notice that only terms of degree 3 and higher remain." }}{PARA 0 "" 0 "" {TEXT 585 93 "_________________ ______________________________________________________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 485 11 "Definiti on:" }{TEXT -1 20 " A bilinear form " }{XPPEDIT 489 1 "B" "I\"BG6\" " }{TEXT -1 8 " on " }{XPPEDIT 487 1 "V*` x `*V" "*(%\"VG\"\"\"%$~ x~GF$F#F$" }{TEXT -1 17 " is said to be " }{TEXT 486 11 " symmetric " }{TEXT -1 40 " if \n\n " }{XPPEDIT 488 1 "B(u,v)=B(v,u)" "/-%\"BG6$%\"uG%\"vG-F$6$F'F&" }{TEXT -1 13 " \+ \n\nfor all " }{XPPEDIT 490 1 "u" "I\"uG6\"" }{TEXT -1 7 " and " } {XPPEDIT 491 1 "v" "I\"vG6\"" }{TEXT -1 5 " in " }{XPPEDIT 492 1 "V" "I\"VG6\"" }{TEXT 493 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 494 8 "Theorem:" }{TEXT -1 10 " \n If " }{XPPEDIT 495 1 "A" "I\"AG6\"" }{TEXT -1 51 " is a sym metric matrix then the bilinear form " }{XPPEDIT 496 1 "B(x,y)=x^t* ``*A*``*y" "/-%\"BG6$%\"xG%\"yG*,)F&%\"tG\"\"\"%!GF+%\"AGF+F,F+F'F+" } {TEXT -1 16 " is symmetric" }{TEXT 497 1 "." }{TEXT -1 8 " \n\n \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 498 6 "Proo f:" }{TEXT -1 11 " Because " }{XPPEDIT 499 1 "B(x,y)=x^t*``*A*``*y" "/-%\"BG6$%\"xG%\"yG*,)F&%\"tG\"\"\"%!GF+%\"AGF+F,F+F'F+" }{TEXT -1 61 " is a scalar we have \n \n " } {XPPEDIT 500 1 "B(x,y)=B(x,y)^t" "/-%\"BG6$%\"xG%\"yG)-F$6$F&F'%\"tG" }{TEXT 508 1 "." }{TEXT -1 23 " \n\nTherefore " }{XPPEDIT 501 1 "B(x,y)=y^t*``*A^t*``*x" "/-%\"BG6$%\"xG%\"yG*,)F'%\"tG\"\"\"%!G F+)%\"AGF*F+F,F+F&F+" }{TEXT 502 1 "." }{TEXT -1 15 " \n\nBecause \+ " }{XPPEDIT 505 1 "A" "I\"AG6\"" }{TEXT -1 57 " is symmetric it foll ows that \n\n " }{XPPEDIT 503 1 "B(x,y)=y^t*``*A *``*x" "/-%\"BG6$%\"xG%\"yG*,)F'%\"tG\"\"\"%!GF+%\"AGF+F,F+F&F+" } {TEXT 504 1 "." }{TEXT -1 46 " \n\nBut this last expression is prec isely " }{XPPEDIT 506 1 "B(y,x)" "-%\"BG6$%\"yG%\"xG" }{TEXT 507 1 " ." }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "4. Inner Product Spaces" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 509 11 "Definition:" }{TEXT -1 20 " A bilinear form " }{XPPEDIT 513 1 " B" "I\"BG6\"" }{TEXT -1 8 " on " }{XPPEDIT 511 1 "V*` x `*V" "*(% \"VG\"\"\"%$~x~GF$F#F$" }{TEXT -1 17 " is said to be " }{TEXT 510 17 " inner product " }{TEXT -1 12 " if \n\n1) " }{XPPEDIT 518 1 " B" "I\"BG6\"" }{TEXT -1 15 " is symmetric" }{TEXT 519 1 ":" }{TEXT -1 7 " " }{XPPEDIT 512 1 "B(u,v)=B(v,u)" "/-%\"BG6$%\"uG%\"vG-F$ 6$F'F&" }{TEXT -1 14 " for all " }{XPPEDIT 514 1 "u" "I\"uG6\"" } {TEXT -1 7 " and " }{XPPEDIT 515 1 "v" "I\"vG6\"" }{TEXT -1 5 " in \+ " }{XPPEDIT 516 1 "V" "I\"VG6\"" }{TEXT 517 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "2) " }{XPPEDIT 520 1 "B" "I\"BG6\"" }{TEXT -1 14 " is \+ positive" }{TEXT 525 1 ":" }{TEXT -1 7 " " }{XPPEDIT 521 1 "B(u, u)>=0" "1\"\"!-%\"BG6$%\"uGF'" }{TEXT -1 14 " for all " } {XPPEDIT 522 1 "u" "I\"uG6\"" }{TEXT -1 5 " in " }{XPPEDIT 523 1 "V" "I\"VG6\"" }{TEXT 524 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "3) " } {XPPEDIT 526 1 "B" "I\"BG6\"" }{TEXT -1 14 " is definite" }{TEXT 530 1 ":" }{TEXT -1 7 " " }{XPPEDIT 527 1 "B(u,u)=0" "/-%\"BG6$% \"uGF&\"\"!" }{TEXT -1 13 " only if " }{XPPEDIT 528 1 "u=0" "/%\"u G\"\"!" }{TEXT 529 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "If " }{XPPEDIT 549 1 "B" "I\"BG6\"" }{TEXT -1 27 " \+ is an inner product on " }{XPPEDIT 548 1 "V*` x `*V" "*(%\"VG\"\"\" %$~x~GF$F#F$" }{TEXT -1 31 " then we say that the pair " } {XPPEDIT 550 1 "[V,B]" "7$%\"VG%\"BG" }{TEXT -1 9 " is an " }{TEXT 547 22 " inner product space" }{TEXT 551 1 "." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 534 10 "Example 1:" } {TEXT -1 8 " Let " }{XPPEDIT 531 1 "A" "I\"AG6\"" }{TEXT -1 12 " \+ be the " }{XPPEDIT 532 1 "n*``*x*``*n" "*,%\"nG\"\"\"%!GF$%\"xGF$F%F $F#F$" }{TEXT -1 74 " diagonal matrix:\n\n \+ " }{XPPEDIT 533 1 "MATRIX([ [d[1],0,0,0,0,0 ],[0,d[2],0,0,0,0],[0,0,`.`,0,0,0],[0,0,0,`.`,0,0],[0,0,0,0,`.`,0],[0, 0,0,0,0,d[n]] ])" "-%'MATRIXG6#7(7(&%\"dG6#\"\"\"\"\"!F+F+F+F+7( F+&F(6#\"\"#F+F+F+F+7(F+F+%\".GF+F+F+7(F+F+F+F1F+F+7(F+F+F+F+F1F+7(F+F +F+F+F+&F(6#%\"nG" }{TEXT -1 4 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The bilinear form " }{XPPEDIT 535 1 " B(x,y) = x^t*``*A*``*y" "/-%\"BG6$%\"xG%\"yG*,)F&%\"tG\"\"\"%!GF+%\"AG F+F,F+F'F+" }{TEXT -1 42 " is an inner product if and only if " }{XPPEDIT 537 1 "d[i]>0" "2\"\"!&%\"dG6#%\"iG" }{TEXT -1 15 " for e ach " }{XPPEDIT 536 1 "i=1..n" "/%\"iG;\"\"\"%\"nG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 624 9 "Notation:" } {TEXT -1 47 " We often denote a fixed inner product by " } {XPPEDIT 626 1 "[v[1],v[2]]->" ":6#7$&%\"vG6#\"\"\"&F&6#\" \"#7\"6$%)operatorG%&arrowG6\"-%-anglebracketG6$&F&6#F(&F&6#F+F0F0" } {TEXT 625 18 ".\n " }{TEXT -1 97 " We will often use Kr onecker delta notation for the values 1 and 0 that an inner product ma y take" }{TEXT 628 1 ":" }{TEXT -1 53 "\n\n \+ " }{XPPEDIT 627 1 "delta [i,j] = PIECEWISE([1, i = j],[0, i <> j])" "/&%&deltaG6$%\"iG%\"jG-%*PIECEWISEG6$7$\"\"\"/F &F'7$\"\"!0F&F'" }{TEXT -1 12 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 " The Krone cker delta notation is used in many other situations as well." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 540 10 "Example 2:" }{TEXT -1 7 " Let " }{XPPEDIT 543 1 " R[n]*`[x]`" "*&&%\"RG6#%\"nG\"\"\"%$[x]GF'" }{TEXT -1 74 " denote th e space of polynomials with real coefficients, indeterminate " } {XPPEDIT 544 1 "x" "I\"xG6\"" }{TEXT -1 37 ", and degree less than or equal to " }{XPPEDIT 542 1 "n" "I\"nG6\"" }{TEXT -1 9 ". Let " } {XPPEDIT 538 1 "w" "I\"wG6\"" }{TEXT -1 49 " be a positive function on the real interval " }{XPPEDIT 541 1 "[a,b]" "7$%\"aG%\"bG" } {TEXT -1 48 ". Then\n\n " } {XPPEDIT 539 1 " =Int(p(x)*q(x)*w(x),x=a..b)" "/-%-anglebracketG6 $%\"pG%\"qG-%$IntG6$*(-F&6#%\"xG\"\"\"-F'6#F.F/-%\"wG6#F.F//F.;%\"aG% \"bG" }{TEXT -1 32 " \n\nis an inner product on " }{XPPEDIT 545 1 "R[n]*`[x]`" "*&&%\"RG6#%\"nG\"\"\"%$[x]GF'" }{TEXT 546 1 "." } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The positive function \+ " }{XPPEDIT 638 1 "w" "I\"wG6\"" }{TEXT -1 17 " is called the " } {TEXT 637 15 "weight function" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 590 93 "__________________________________________________________________ ___________________________" }}{PARA 0 "" 0 "" {TEXT 587 6 "Maple:" } {TEXT -1 79 " Define an inner product on polynomials by \n\n\n \+ " }{XPPEDIT 631 1 " =Int(p(x)*q(x),x=-1. .1)" "/-%-anglebracketG6$%\"pG%\"qG-%$IntG6$*&-F&6#%\"xG\"\"\"-F'6#F.F //F.;,$F/!\"\"F/" }{TEXT -1 9 "\n\n Use " }{TEXT 589 5 "Maple" } {TEXT -1 25 " to calculate a basis " }{XPPEDIT 596 1 "[v[0],v[1],v[ 2]]" "7%&%\"vG6#\"\"!&F$6#\"\"\"&F$6#\"\"#" }{TEXT -1 8 " of " } {XPPEDIT 597 1 "R[2]*`[x]`" "*&&%\"RG6#\"\"#\"\"\"%$[x]GF'" }{TEXT -1 15 " such that " }{XPPEDIT 598 1 "< v[ i ] , v[ j ] > = delta[i,j " "/-%-anglebracketG6$&%\"vG6#%\"iG&F'6#%\"jG&%&deltaG6$F)F," }{TEXT -1 9 " and " }{XPPEDIT 632 1 "deg(v[i])=i" "/-%$degG6#&%\"vG6#%\"i GF)" }{TEXT 629 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 588 9 "Solution:" }{TEXT -1 4 " \n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Re-execute the homebrew \+ " }{TEXT 592 5 "Maple" }{TEXT -1 12 " function " }{TEXT 593 14 " bil inearForm " }{TEXT -1 46 " (found in the preceding section).\n\nTry ou t " }{TEXT 633 14 " bilinearForm " }{TEXT -1 32 " on a generic par ameter list:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "bilinearFo rm(p,q,w,a..b); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*(-%\"pG6 #%\"xG\"\"\"-%\"qGF)F+-%\"wGF)F+/F*;%\"aG%\"bG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Now let us use the interval " }{XPPEDIT 635 1 "[-1 , 1 ]" "7$, $\"\"\"!\"\"F$" }{TEXT -1 14 " and weight " }{MPLTEXT 1 0 6 "w(x)=1 " }{TEXT -1 3 ":\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w := x -> 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "bilinearForm(p, q, w, -1 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&-%\"pG6 #%\"xG\"\"\"-%\"qGF)F+/F*;!\"\"F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 639 1 "v[0]" "&%\"vG6# \"\"!" }{TEXT -1 21 " is a constant and " }{XPPEDIT 640 1 " " "-%-anglebracketG6$&%\"vG6#\"\"!&F&6#F(" }{TEXT -1 9 " equals " }{XPPEDIT 641 1 "1 " "\"\"\"" }{TEXT -1 26 " we find it as follows:\n \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v[0] := x -> a;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"!:6#%\"xG6\"6$%)operatorG %&arrowGF+%\"aGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqn \+ := bilinearForm(v[0], v[0], w, -1 .. 1) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*$%\"aG\"\"#F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(eqn,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $,$*$\"\"##\"\"\"F%F&,$F$#!\"\"F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a := 1/2*2^(1/2); v[0](x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,$*$\"\"##\"\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"##\"\"\"F%F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Next, let " }{XPPEDIT 642 1 "v[1]=b+c*x" "/&%\"vG6#\"\"\",&%\"bGF&*&%\"cGF&%\"xGF&F&" } {TEXT -1 31 " . We use the two equations " }{XPPEDIT 643 1 " < v[ 1 ] , v[ 1 ] > = 1" "/-%-anglebracketG6$&%\"vG6#\"\"\"&F'6#F)F)" } {TEXT -1 7 " and " }{XPPEDIT 644 1 " < v[ 1 ] , v[ 1 ] > = 0" "/-%-a nglebracketG6$&%\"vG6#\"\"\"&F'6#F)\"\"!" }{TEXT -1 34 " to solve for the two constants " }{XPPEDIT 645 1 "b" "I\"bG6\"" }{TEXT -1 7 " an d " }{XPPEDIT 646 1 "c" "I\"cG6\"" }{TEXT -1 1 " " }{TEXT 647 1 ":" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "v[1] := x -> b+c*x:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 " eqn1 := bilinearForm(v[1],v[1],w,-1..1) = 1;\neqn2 := bilinearForm(v[0 ],v[1],w,-1..1) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*&, **$%\"bG\"\"$\"\"\"*&F*\"\"#%\"cGF,F+*&F*F,F/F.F+*$F/F+F,F,F/!\"\"#F,F +*&,*F)F2F-F+F0!\"$F1F,F,F/F2F3F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%eqn2G/*&\"\"##\"\"\"F'%\"bGF)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solnSet := solve( \{eqn1,eqn2\},\{b,c\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(solnSetG<$/%\"bG\"\"!/%\"cG-%'RootOfG6#,& !\"$\"\"\"*$%#_ZG\"\"#F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "allvalues(solnSet); # Forces the evaluation of 'RootOf'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"bG\"\"!/%\"cG,$*$\"\"'#\"\"\"\"\"#F,<$F$/ F(,$F*#!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b := 0:\n c := 1/2*6^(1/2):\nv[1](x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\" \"'#\"\"\"\"\"#%\"xGF'F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Next, let " }{XPPEDIT 648 1 "v[2]=d+e*x+f*x^2" "/&%\"v G6#\"\"#,(%\"dG\"\"\"*&%\"eGF)%\"xGF)F)*&%\"fGF)*$F,F&F)F)" }{TEXT -1 33 " . We use the three equations " }{XPPEDIT 649 1 " < v[ 2 ] , v[ 2 ] > = 1" "/-%-anglebracketG6$&%\"vG6#\"\"#&F'6#F)\"\"\"" }{TEXT -1 5 " , " }{XPPEDIT 654 1 " < v[ 0 ] , v[ 2 ] > = 0" "/-%-anglebra cketG6$&%\"vG6#\"\"!&F'6#\"\"#F)" }{TEXT -1 9 " , and " }{XPPEDIT 650 1 "< v[ 1 ] , v[ 2 ] > = 0" "/-%-anglebracketG6$&%\"vG6#\"\"\"&F'6 #\"\"#\"\"!" }{TEXT -1 36 " to solve for the three constants " } {XPPEDIT 651 1 "d" "I\"dG6\"" }{TEXT -1 6 " , " }{XPPEDIT 655 1 "e " "I\"eG6\"" }{TEXT -1 8 ", and " }{XPPEDIT 652 1 "f" "I\"fG6\"" } {TEXT -1 1 " " }{TEXT 653 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v[2] := x -> d + e*x + f*x^2 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "eqn1 := bilinearForm( v[2],v[2],w,-1..1) = 1;\neqn2 := bilinearForm(v[0],v[2],w,-1..1) = 0; \neqn3 := bilinearForm(v[1],v[2],w,-1..1) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,**$%\"fG\"\"##F)\"\"&*&%\"dG\"\"\"F(F.#\"\"% \"\"$*$%\"eGF)#F)F1*$F-F)F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq n2G/,&*&\"\"##\"\"\"F(%\"dGF*F**&F(F)%\"fGF*#F*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,$*&\"\"'#\"\"\"\"\"#%\"eGF*#F*\"\"$\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solnSet := solve( \{ eqn1,eqn2,eqn3\},\{d,e,f\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sol nSetG<%/%\"eG\"\"!/%\"dG,$-%'RootOfG6#,&*$%#_ZG\"\"#F2!\"&\"\"\"#F4F2/ %\"fG,$F,#!\"$F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "allvalu es(solnSet); # Forces the evaluation of 'RootOf'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<%/%\"eG\"\"!/%\"dG,$*$\"#5#\"\"\"\"\"##F-\"\"%/%\"fG,$ F*#!\"$F0<%F$/F(,$F*#!\"\"F0/F2,$F*#\"\"$F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "d := -1/4*10^(1/2):\ne := 0:\nf := 3/4*10^(1/2); \+ \nv[2](x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$*$\"#5#\"\"\"\" \"##\"\"$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"#5#\"\"\"\"\"# #!\"\"\"\"%*&F%F&%\"xGF(#\"\"$F+" }}}{PARA 0 "" 0 "" {TEXT -1 15 "\nHe re they are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "'v[0](x)' = v[0](x);\n'v[1](x)' = v[1](x);\n'v[2]( x)' = v[2](x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"vG6#\"\"!6#%\" xG,$*$\"\"##\"\"\"F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"vG6#\" \"\"6#%\"xG,$*&\"\"'#F(\"\"#F*F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-&%\"vG6#\"\"#6#%\"xG,&*$\"#5#\"\"\"F(#!\"\"\"\"%*&F-F.F*F(#\"\"$F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "As a check:\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "for i from 0 to 2 do\n for j from 0 to 2 do\n \+ print( 'bilinearForm'(v[i],v[j],w,-1..1) = bilinearForm(v[i],v[j], w,-1..1));\n od;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinea rFormG6&&%\"vG6#\"\"!F'\"\"\";!\"\"F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFormG6&&%\"vG6#\"\"!&F(6#\"\"\"F-;!\"\"F-F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFormG6&&%\"vG6#\"\"!&F(6# \"\"#\"\"\";!\"\"F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearF ormG6&&%\"vG6#\"\"\"&F(6#\"\"!F*;!\"\"F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFormG6&&%\"vG6#\"\"\"F'F*;!\"\"F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFormG6&&%\"vG6#\"\"\"&F(6# \"\"#F*;!\"\"F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFo rmG6&&%\"vG6#\"\"#&F(6#\"\"!\"\"\";!\"\"F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFormG6&&%\"vG6#\"\"#&F(6#\"\"\"F-;!\"\"F-\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%-bilinearFormG6&&%\"vG6#\"\" #F'\"\"\";!\"\"F+F+" }}}{PARA 0 "" 0 "" {TEXT 594 93 "________________ ______________________________________________________________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "5. Orthogonality" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 656 11 "Definition:" }{TEXT -1 24 " We say two vectors \+ " }{XPPEDIT 667 1 "u" "I\"uG6\"" }{TEXT -1 7 " and " }{XPPEDIT 668 1 "v" "I\"vG6\"" }{TEXT -1 29 " in an inner product space " } {XPPEDIT 669 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$%$~.~GF' " }{TEXT -1 8 " are " }{TEXT 657 12 " orthogonal" }{TEXT -1 43 " \+ if \n\n " }{XPPEDIT 659 1 "< `u , \+ v ` > *`= 0 `" "*&-%-anglebracketG6#%(u~~,~v~G\"\"\"%%=~0~GF'" }{TEXT 664 3 ".\n\n" }}{PARA 0 "" 0 "" {TEXT -1 30 "Notice that scalar multip les " }{XPPEDIT 724 1 "a*u" "*&%\"aG\"\"\"%\"uGF$" }{TEXT -1 7 " and " }{XPPEDIT 725 1 "b*v" "*&%\"bG\"\"\"%\"vGF$" }{TEXT -1 26 " of or thogonal vectors " }{XPPEDIT 726 1 "u" "I\"uG6\"" }{TEXT -1 7 " and " }{XPPEDIT 727 1 "v" "I\"vG6\"" }{TEXT -1 22 " are also orthogonal ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 670 8 "Theo rem:" }{TEXT -1 10 " If " }{XPPEDIT 671 1 "\{ u[1],u[2],`...`,u[ n] \}" "<&&%\"uG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 64 " is a collection of mutually orthogonal nonzero vectors then " } {XPPEDIT 673 1 "\{ u[1],u[2],`...`,u[n] \}" "<&&%\"uG6#\"\"\"&F$6#\"\" #%$...G&F$6#%\"nG" }{TEXT -1 35 " is an independent set of vectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 674 6 "Proof: " }{TEXT -1 18 " Suppose that " }{XPPEDIT 675 1 "c[1]*u[1]+c[2]*u[ 2]+`...`+c[i]*u[i]+`...`+c[n]*u[n] =0" "/,.*&&%\"cG6#\"\"\"F(&%\"uG6#F (F(F(*&&F&6#\"\"#F(&F*6#F/F(F(%$...GF(*&&F&6#%\"iGF(&F*6#F6F(F(F2F(*&& F&6#%\"nGF(&F*6#F *`= ` *0" "*(-%-anglebrack etG6$,.*&&%\"cG6#\"\"\"F+&%\"uG6#F+F+F+*&&F)6#\"\"#F+&F-6#F2F+F+%$...G F+*&&F)6#%\"iGF+&F-6#F9F+F+F5F+*&&F)6#%\"nGF+&F-6#F?F+F+&F-6#F9F+%#=~G F+\"\"!F+" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Using linearity this becomes\n " }{XPPEDIT 680 1 "c[1]*< u[ 1 ] , u[ i ] > + c[2]*< u[ 2 ] , u[ i ] > \+ + `... `+c[i]*< u[ i ] , u[ i ] >+`...`+c[n]*< u[n] , u[i] > = 0" " /,.*&&%\"cG6#\"\"\"F(-%-anglebracketG6$&%\"uG6#F(&F-6#%\"iGF(F(*&&F&6# \"\"#F(-F*6$&F-6#F5&F-6#F1F(F(%%...~GF(*&&F&6#F1F(-F*6$&F-6#F1&F-6#F1F (F(%$...GF(*&&F&6#%\"nGF(-F*6$&F-6#FJ&F-6#F1F(F(\"\"!" }{TEXT -1 1 " \+ " }{TEXT 681 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "Using the the assumed orthogonality this simplifies to\n" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 684 1 "c[i]*< u[ i ] , u[ i ] > = 0" "/*&&%\"cG6#%\"iG\"\"\"-%-anglebrack etG6$&%\"uG6#F'&F-6#F'F(\"\"!" }{TEXT -1 1 " " }{TEXT 685 1 "." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 686 1 "u[ i ] " "&%\"uG6#%\"iG" }{TEXT -1 21 " is nozero we have " }{XPPEDIT 688 1 "< u[ i ] , u[ i ] > > \+ 0" "2\"\"!-%-anglebracketG6$&%\"uG6#%\"iG&F(6#F*" }{TEXT 689 1 "." } {TEXT -1 15 " Therefore " }{XPPEDIT 690 1 "c[i] = 0" "/&%\"cG6#% \"iG\"\"!" }{TEXT -1 1 " " }{TEXT 691 1 "." }{TEXT -1 98 " Since thi s was an arbitrary coefficient in the dependence relation linear indep endence follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 917 11 "Definition:" }{TEXT -1 27 " We say that a vector " } {XPPEDIT 921 1 "u" "I\"uG6\"" }{TEXT -1 30 " in an inner product spa ce " }{XPPEDIT 923 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$% $~.~GF'" }{TEXT -1 7 " is " }{TEXT 918 12 " normalized" }{TEXT -1 44 " if \n\n " }{XPPEDIT 919 1 "< u , u > = 1" "/-%-anglebracketG6$%\"uGF&\"\"\"" }{TEXT 920 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 924 8 "Theorem:" }{TEXT -1 24 " Every nonzero \+ vector " }{XPPEDIT 926 1 "u" "I\"uG6\"" }{TEXT -1 29 " in an inner pr oduct space " }{XPPEDIT 927 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebr acketG6$%$~.~GF'" }{TEXT -1 42 " is proportional to a normalized vec tor." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 928 6 "P roof:" }{TEXT -1 18 " The vector " }{XPPEDIT 929 1 "u/sqrt(< u \+ , u >)" "*&%\"uG\"\"\"-%%sqrtG6#-%-anglebracketG6$F#F#!\"\"" }{TEXT -1 16 " is normalized." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 930 11 "Definition:" }{TEXT -1 36 " We say that a set of vec tors " }{XPPEDIT 931 1 "\{ u[1],u[2],`...`,u[n] \}" "<&&%\"uG6#\"\" \"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 9 " is an " }{TEXT 941 15 "o rthonormal set" }{TEXT -1 7 " if " }{XPPEDIT 932 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$%$~.~GF'" }{TEXT -1 61 " . Then we ca n find a set of mutually orthogonal vectors " }{XPPEDIT 933 1 "\{ v[ 1],v[2],`...`,v[n] \}" "<&&%\"vG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" } {TEXT -1 7 " with " }{XPPEDIT 940 1 " = delta[i,j]" "/-%- anglebracketG6$&%\"uG6#%\"iG&F'6#%\"jG&%&deltaG6$F)F," }{TEXT -1 14 " \+ for each " }{XPPEDIT 939 1 "i" "I\"iG6\"" }{TEXT -1 27 " and such that for each " }{XPPEDIT 934 1 "j" "I\"jG6\"" }{TEXT -1 16 " in t he range " }{XPPEDIT 935 1 "1..n" ";\"\"\"%\"nG" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 692 8 "Theorem:" } {TEXT -1 2 " " }{TEXT 702 32 "(Gram-Schmidt Orthogonalization)" } {TEXT -1 18 " \nSuppose that " }{XPPEDIT 694 1 "\{ u[1],u[2],`...`, u[n] \}" "<&&%\"uG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"nG" }{TEXT -1 62 " \+ is an independent set of vectors in an inner product space " } {XPPEDIT 695 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$%$~.~GF' " }{TEXT -1 45 " . Then we can find an orthonormal set " } {XPPEDIT 696 1 "\{ v[1],v[2],`...`,v[n] \}" "<&&%\"vG6#\"\"\"&F$6#\"\" #%$...G&F$6#%\"nG" }{TEXT -1 25 " such that for each " }{XPPEDIT 697 1 "j" "I\"jG6\"" }{TEXT -1 16 " in the range " }{XPPEDIT 698 1 " 1..n" ";\"\"\"%\"nG" }{TEXT -1 14 " the subset " }{XPPEDIT 699 1 "\{ v[1],v[2],`...`,v[ j ] \}" "<&&%\"vG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"j G" }{TEXT -1 36 " spans the same subspace as does " }{XPPEDIT 700 1 "\{ u[1],u[2],`...`,u[ j ] \}" "<&&%\"uG6#\"\"\"&F$6#\"\"#%$...G&F$6 #%\"jG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 701 6 "Proo f:" }{TEXT -1 10 " Let " }{XPPEDIT 706 1 "U[i]" "&%\"UG6#%\"iG" } {TEXT -1 33 " denote the space spanned by " }{XPPEDIT 705 1 "\{ u[ 1],u[2],`...`,u[i] \}" "<&&%\"uG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"iG" } {TEXT -1 12 ". \n\nLet " }{XPPEDIT 703 1 "v[1] = u[1]/sqrt()" "/&%\"vG6#\"\"\"*&&%\"uG6#F&F&-%%sqrtG6#-%-anglebracketG6$&F)6# F&&F)6#F&!\"\"" }{TEXT -1 1 " " }{TEXT 736 1 "." }{TEXT -1 24 " Clea rly the span of " }{XPPEDIT 719 1 "\{ v[1] \}" "<#&%\"vG6#\"\"\"" } {TEXT -1 34 " is is the same as the span of " }{XPPEDIT 720 1 "\{ u [1] \}" "<#&%\"uG6#\"\"\"" }{TEXT -1 20 " .\nFurthermore, " } {XPPEDIT 721 1 " = /(sqrt() ^2" "/-%-anglebracketG6$&%\"vG6#\"\"\"&F'6#\"\"#*&-F$6$&%\"uG6#F)&F16# F)F)*$-%%sqrtG6#-F$6$&F16#F)&F16#F)F,!\"\"" }{TEXT -1 20 " \nwhich sho ws that " }{XPPEDIT 723 1 " = 1" "/-%-anglebrack etG6$&%\"vG6#\"\"\"&F'6#F)F)" }{TEXT 722 1 "." }{TEXT -1 12 " \n\n\nLe t " }{XPPEDIT 704 1 "w[2] = u[2] - (< v[ 1 ] , u[ 2] > *v[1]" "/&% \"wG6#\"\"#,&&%\"uG6#F&\"\"\"*&-%-anglebracketG6$&%\"vG6#F+&F)6#F&F+&F 16#F+F+!\"\"" }{TEXT 710 2 ". " }{TEXT -1 114 " This vector cannot be \+ the zero vector: if it were then we would have a nontrivial dependence relation between " }{XPPEDIT 728 1 "u[1]" "&%\"uG6#\"\"\"" }{TEXT -1 7 " and " }{XPPEDIT 729 1 "u[2]" "&%\"uG6#\"\"#" }{TEXT -1 19 " \+ . Consequently " }{XPPEDIT 731 1 " > 0" "2\"\"!-% -anglebracketG6$&%\"wG6#\"\"#&F(6#F*" }{TEXT 730 1 "." }{TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Obs erve that " }{XPPEDIT 712 1 " = - < v[ 1 ] , u[ 2] > *" "/-%-anglebracketG6$&%\"vG6#\"\"\"&%\" wG6#\"\"#,&-F$6$&F'6#F)&%\"uG6#F-F)*&-F$6$&F'6#F)&F46#F-F)-F$6$&F'6#F) &F'6#F)F)!\"\"" }{TEXT 711 3 ", " }{TEXT -1 19 " which shows that " }{XPPEDIT 732 1 " = 0" "/-%-anglebracketG6$&%\"vG6 #\"\"\"&%\"wG6#\"\"#\"\"!" }{TEXT -1 11 " since " }{XPPEDIT 734 1 " = 1" "/-%-anglebracketG6$&%\"vG6#\"\"\"&F'6#F)F )" }{TEXT 733 1 "." }{TEXT -1 11 " Let " }{XPPEDIT 735 1 "v[2] = w[2]/sqrt()" "/&%\"vG6#\"\"#*&&%\"wG6#F&\"\"\"-%%sqrtG6#-% -anglebracketG6$&F)6#F&&F)6#F&!\"\"" }{TEXT -1 1 " " }{TEXT 737 1 "." }{TEXT -1 12 " Then " }{XPPEDIT 738 1 " = 1" "/-%-ang lebracketG6$&%\"vG6#\"\"#&F'6#F)\"\"\"" }{TEXT -1 8 " and " } {XPPEDIT 739 1 " = 0" "/-%-anglebracketG6$&%\"vG6#\"\"#&F'6 #\"\"\"\"\"!" }{TEXT -1 1 " " }{TEXT 741 1 "." }{TEXT -1 23 " \nAlso\n \n " }{XPPEDIT 742 1 "b*sqrt() )*v[2] = b*u[2 ] - b**v[1]" "/*(%\"bG\"\"\"-%%sqrtG6#-%-anglebracketG6$&% \"wG6#\"\"#&F-6#F/F%&%\"vG6#F/F%,&*&F$F%&%\"uG6#F/F%F%*(F$F%-F*6$&F36# F%&F86#F/F%&F36#F%F%!\"\"" }{TEXT -1 10 " \n\n\nor" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " " } {XPPEDIT 743 1 "b*sqrt() )*v[2] = b*u[2] - b**u[ 1]/sqrt(" "/*(%\"bG\"\"\"-%%sqrtG6#-%-anglebracketG6$&%\"wG 6#\"\"#&F-6#F/F%&%\"vG6#F/F%,&*&F$F%&%\"uG6#F/F%F%**F$F%-F*6$&F36#F%&F 86#F/F%&F86#F%F%-F'6#-F*6$&F86#F%&F86#F%!\"\"FK" }{TEXT -1 1 " " } {TEXT 744 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "It follows that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 " \n " }{XPPEDIT 713 1 "a*u[1] + b*u[2] =(a+b*((< v[ 1 ] , u[ 2] >/sqrt() )))*u[1]+b*sqrt( )*v[2]" "/,&*&%\"aG\"\"\"&%\"uG6#F&F&F&*&%\"bGF&&F(6#\"\"#F &F&,&*&,&F%F&*&F+F&*&-%-anglebracketG6$&%\"vG6#F&&F(6#F.F&-%%sqrtG6#-F 56$&F(6#F&&F(6#F&!\"\"F&F&F&&F(6#F&F&F&*(F+F&-F=6#-F56$&%\"wG6#F.&FN6# F.F&&F86#F.F&F&" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " \+ " }{XPPEDIT 745 1 "a*u[1] + b*u[2] =(a*sqrt( ) +b*< v[ 1 ] , u[ 2] > )*v[1] +b*sqrt()*v[2]" "/,&*&%\"aG\"\" \"&%\"uG6#F&F&F&*&%\"bGF&&F(6#\"\"#F&F&,&*&,&*&F%F&-%%sqrtG6#-%-angleb racketG6$&F(6#F&&F(6#F&F&F&*&F+F&-F76$&%\"vG6#F&&F(6#F.F&F&F&&FA6#F&F& F&*(F+F&-F46#-F76$&%\"wG6#F.&FM6#F.F&&FA6#F.F&F&" }{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "which s hows that the span of " }{XPPEDIT 717 1 "\{ v[1],v[2] \}" "<$&%\"vG6# \"\"\"&F$6#\"\"#" }{TEXT -1 34 " is is the same as the span of " } {XPPEDIT 718 1 "\{ u[1],u[2] \}" "<$&%\"uG6#\"\"\"&F$6#\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 19 "In general, once " } {XPPEDIT 707 1 "\{ v[1],v[2],`...`,v[j] \}" "<&&%\"vG6#\"\"\"&F$6#\"\" #%$...G&F$6#%\"jG" }{TEXT -1 36 " have been defined, define\n\n \+ " }{XPPEDIT 708 1 "w[j+1] = u[j+1] - < v[1], u[ j+1 ] >*v[1] - < v[2 ], u[ j+1 ] >*v[2] - `...`- < v[j], u[ j+1 ] >*v[j]" "/&%\"wG6#,&%\" jG\"\"\"F(F(,,&%\"uG6#,&F'F(F(F(F(*&-%-anglebracketG6$&%\"vG6#F(&F+6#, &F'F(F(F(F(&F36#F(F(!\"\"*&-F06$&F36#\"\"#&F+6#,&F'F(F(F(F(&F36#F@F(F: %$...GF:*&-F06$&F36#F'&F+6#,&F'F(F(F(F(&F36#F'F(F:" }{TEXT -1 1 " " } {TEXT 709 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "For any " }{XPPEDIT 746 1 "i" "I\"iG6\"" }{TEXT -1 7 " in " }{XPPEDIT 747 1 "1..j" ";\"\"\"%\"jG" }{TEXT -1 11 " we have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 749 1 " = - < v[1], u[ j+1 ] >* - < v[2], u[ j+1 ] >* - `...`- *-`...`-< v[j], u[ j+1 ] >*v [j]" "/-%-anglebracketG6$&%\"vG6#%\"iG&%\"wG6#,&%\"jG\"\"\"F/F/,0-F$6$ &F'6#F)&%\"uG6#,&F.F/F/F/F/*&-F$6$&F'6#F/&F66#,&F.F/F/F/F/-F$6$&F'6#F) &F'6#F/F/!\"\"*&-F$6$&F'6#\"\"#&F66#,&F.F/F/F/F/-F$6$&F'6#F)&F'6#FMF/F G%$...GFG*&-F$6$&F'6#F)&F66#,&F.F/F/F/F/-F$6$&F'6#F)&F'6#F)F/FGFWFG*&- F$6$&F'6#F.&F66#,&F.F/F/F/F/&F'6#F.F/FG" }{TEXT -1 1 " " }{TEXT 750 1 "." }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 36 "By orthogonality this simplifies to " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ " }{XPPEDIT 751 1 " = -*" "/-%-anglebracketG6$&%\"vG6#%\"iG&%\"wG6#,&%\"jG\"\"\"F/F/,& -F$6$&F'6#F)&%\"uG6#,&F.F/F/F/F/*&-F$6$&F'6#F)&F66#,&F.F/F/F/F/-F$6$&F '6#F)&F'6#F)F/!\"\"" }{TEXT -1 1 " " }{TEXT 754 1 "." }{TEXT -1 6 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "It follows that " }{XPPEDIT 755 1 " = 0" "/-%-anglebrac ketG6$&%\"vG6#%\"iG&%\"wG6#,&%\"jG\"\"\"F/F/\"\"!" }{TEXT -1 10 " si nce " }{XPPEDIT 753 1 " = 1" "/-%-anglebracketG6$&%\"vG6#% \"iG&F'6#F)\"\"\"" }{TEXT 756 1 "." }{TEXT -1 10 " Let " } {XPPEDIT 758 1 "v[j+1] = w[j+1]/sqrt()" "/&%\"vG6#,&%\" jG\"\"\"F(F(*&&%\"wG6#,&F'F(F(F(F(-%%sqrtG6#-%-anglebracketG6$&F+6#,&F 'F(F(F(&F+6#,&F'F(F(F(!\"\"" }{TEXT 757 1 "." }{TEXT -1 8 " Then " } {XPPEDIT 759 1 " = 0" "/-%-anglebracketG6$&%\"vG6#%\"iG &F'6#,&%\"jG\"\"\"F.F.\"\"!" }{TEXT -1 12 " for any " }{XPPEDIT 760 1 "i" "I\"iG6\"" }{TEXT -1 7 " in " }{XPPEDIT 761 1 "1..j" ";\" \"\"%\"jG" }{TEXT -1 19 " . Furthermore " }{XPPEDIT 762 1 " = 1" "/-%-anglebracketG6$&%\"vG6#,&%\"jG\"\"\"F+F+&F'6#,&F*F+ F+F+F+" }{TEXT 763 1 "." }{TEXT -1 39 " Finally, the defining relati on for " }{XPPEDIT 764 1 "w[j+1]" "&%\"wG6#,&%\"jG\"\"\"F'F'" }{TEXT -1 13 " shows that " }{XPPEDIT 765 1 "\{ u[1],u[2],`...`,u[j] ,u[j+1] \}" "<'&%\"uG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"jG&F$6#,&F-F&F&F&" } {TEXT -1 6 " and " }{XPPEDIT 766 1 "\{ v[1],v[2],`...`,v[j] ,w[j+1]\} " "<'&%\"vG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"jG&%\"wG6#,&F-F&F&F&" } {TEXT -1 56 " have the same span. The same is of course true for \+ " }{XPPEDIT 767 1 "\{ v[1],v[2],`...`,v[j] ,v[j+1]\}" "<'&%\"vG6#\"\" \"&F$6#\"\"#%$...G&F$6#%\"jG&F$6#,&F-F&F&F&" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 768 8 "Example:" }{TEXT -1 69 " Apply the Gram-Schmidt O rthogonalization process to the vectors " }{XPPEDIT 769 1 "u[1] = MAT RIX( [ [1],[0],[-1],[1] ] )" "/&%\"uG6#\"\"\"-%'MATRIXG6#7&7#F& 7#\"\"!7#,$F&!\"\"7#F&" }{TEXT -1 5 " , " }{XPPEDIT 770 1 "u[2]=MATR IX( [ [0],[1],[0],[1] ] )" "/&%\"uG6#\"\"#-%'MATRIXG6#7&7#\"\"! 7#\"\"\"7#F,7#F." }{TEXT -1 8 ", and " }{XPPEDIT 771 1 "u[3]=MATRIX( [ [0],[0],[2],[1] ] )" "/&%\"uG6#\"\"$-%'MATRIXG6#7&7#\"\"!7#F ,7#\"\"#7#\"\"\"" }{TEXT -1 56 ". (Use the standard dot product for t he inner product.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 782 9 "Solution:" }{TEXT -1 21 " First calculate " } {XPPEDIT 783 1 "sqrt() = sqrt(3)" "/-%%sqrtG6#-%-anglebrack etG6$&%\"uG6#\"\"\"&F*6#F,-F$6#\"\"$" }{TEXT -1 13 " . Set " } {XPPEDIT 784 1 "v[1] = 3^(-1/2)*MATRIX( [ [1],[0],[-1],[1] ] ) " "/&%\"vG6#\"\"\"*&)\"\"$,$*&F&F&\"\"#!\"\"F-F&-%'MATRIXG6#7&7#F&7#\" \"!7#,$F&F-7#F&F&" }{TEXT -1 11 ". That is," }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 797 1 "v[1] = MATRIX( [ [1/sqrt(3)] , [0],[-1/sqrt(3)] , [1/sqrt(3)] ] \+ )" "/&%\"vG6#\"\"\"-%'MATRIXG6#7&7#*&F&F&-%%sqrtG6#\"\"$!\"\"7#\"\"! 7#,$*&F&F&-F.6#F0F1F17#*&F&F&-F.6#F0F1" }{TEXT -1 2 " " }{TEXT 798 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Next let " }{XPPEDIT 785 1 "w[2] = u[ 2]-*v[1]" "/&%\"wG6#\"\"#,&&%\"uG6#F&\"\"\"*&-%-anglebracke tG6$&%\"vG6#F+&F)6#F&F+&F16#F+F+!\"\"" }{TEXT -1 10 " or " } {XPPEDIT 786 1 "w[2] = MATRIX( [ [0],[1],[0],[1] ] ) - < 3^( -1/2)*MATRIX( [ [1],[0],[-1],[1] ] ) , MATRIX( [ [0],[1], [0],[1] ] )>* 3^(-1/2)*MATRIX( [ [1],[0],[-1],[1] ] )" "/& %\"wG6#\"\"#,&-%'MATRIXG6#7&7#\"\"!7#\"\"\"7#F-7#F/F/*(-%-anglebracket G6$*&)\"\"$,$*&F/F/F&!\"\"F;F/-F)6#7&7#F/7#F-7#,$F/F;7#F/F/-F)6#7&7#F- 7#F/7#F-7#F/F/)F8,$*&F/F/F&F;F;F/-F)6#7&7#F/7#F-7#,$F/F;7#F/F/F;" } {TEXT -1 13 " .\n\nThus, " }{XPPEDIT 787 1 "w[2] = MATRIX( [ [0 ],[1],[0],[1] ] ) - 3^(-1 )*MATRIX( [ [1],[0],[-1],[1] ] ) " "/&%\"wG6#\"\"#,&-%'MATRIXG6#7&7#\"\"!7#\"\"\"7#F-7#F/F/*&)\"\"$,$F/ !\"\"F/-F)6#7&7#F/7#F-7#,$F/F67#F/F/F6" }{TEXT -1 10 " or " } {XPPEDIT 788 1 "w[2] = MATRIX( [ [-1/3],[1],[1/3],[2/3] ] )" "/&%\"wG6#\"\"#-%'MATRIXG6#7&7#,$*&\"\"\"F.\"\"$!\"\"F07#F.7#*&F.F.F/F 07#*&F&F.F/F0" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 " Calculate " }{XPPEDIT 789 1 "sqrt( ) = sqrt(1/9+1+1/9+4/9)" "/-%%sqrtG6#-%-anglebracketG6$&%\"wG 6#\"\"#&F*6#F,-F$6#,**&\"\"\"F3\"\"*!\"\"F3F3F3*&F3F3F4F5F3*&\"\"%F3F4 F5F3" }{TEXT -1 15 " . That is, " }{XPPEDIT 791 1 "sqrt( ) = sqrt(15)/3" "/-%%sqrtG6#-%-anglebracketG6$&%\"wG6#\"\"#&F*6#F,* &-F$6#\"#:\"\"\"\"\"$!\"\"" }{TEXT -1 8 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Set " }{XPPEDIT 792 1 "v[2] = w[2]/sqrt( ) " "/& %\"vG6#\"\"#*&&%\"wG6#F&\"\"\"-%%sqrtG6#-%-anglebracketG6$&F)6#F&&F)6# F&!\"\"" }{TEXT -1 10 " or " }{XPPEDIT 793 1 "v[2] = MATRIX( \+ [ [-1/sqrt(15)],[3/sqrt(15)],[1/sqrt(15)],[2/sqrt(15)] ] )" "/&% \"vG6#\"\"#-%'MATRIXG6#7&7#,$*&\"\"\"F.-%%sqrtG6#\"#:!\"\"F37#*&\"\"$F .-F06#F2F37#*&F.F.-F06#F2F37#*&F&F.-F06#F2F3" }{TEXT -1 1 " " }{TEXT 794 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Now let " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }{XPPEDIT 795 1 "w[3] = u[3]-*v[1] - *v[2]" "/&%\"wG6#\"\"$,(&%\"uG6#F&\"\"\"*&-%-anglebracketG6$&%\"v G6#F+&F)6#F&F+&F16#F+F+!\"\"*&-F.6$&F16#\"\"#&F)6#F&F+&F16#F=F+F7" } {TEXT -1 5 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Thus " }}{PARA 0 "" 0 "" {TEXT -1 8 " " } {XPPEDIT 796 1 "w[3] = MATRIX( [ [0],[0],[2],[1] ] ) - < \+ MATRIX( [ [1/sqrt(3)] , [0],[-1/sqrt(3)] , [1/sqrt(3)] ] ) , MATRIX( [ [0],[0],[2],[1] ] ) >*MATRIX( [ [1/sqrt(3 )] , [0],[-1/sqrt(3)] , [1/sqrt(3)] ] ) - < MATRIX( [ [-1/sqrt(15)],[3/sqrt(15)],[1/sqrt(15)],[2/sqrt(15)] ] ) , \+ MATRIX( [ [0],[0],[2],[1] ] ) >* MATRIX( [ [-1/sqrt(15) ],[3/sqrt(15)],[1/sqrt(15)],[2/sqrt(15)] ] )" "/&%\"wG6#\"\"$,(-%' MATRIXG6#7&7#\"\"!7#F-7#\"\"#7#\"\"\"F2*&-%-anglebracketG6$-F)6#7&7#*& F2F2-%%sqrtG6#F&!\"\"7#F-7#,$*&F2F2-F=6#F&F?F?7#*&F2F2-F=6#F&F?-F)6#7& 7#F-7#F-7#F07#F2F2-F)6#7&7#*&F2F2-F=6#F&F?7#F-7#,$*&F2F2-F=6#F&F?F?7#* &F2F2-F=6#F&F?F2F?*&-F56$-F)6#7&7#,$*&F2F2-F=6#\"#:F?F?7#*&F&F2-F=6#Fg oF?7#*&F2F2-F=6#FgoF?7#*&F0F2-F=6#FgoF?-F)6#7&7#F-7#F-7#F07#F2F2-F)6#7 &7#,$*&F2F2-F=6#FgoF?F?7#*&F&F2-F=6#FgoF?7#*&F2F2-F=6#FgoF?7#*&F0F2-F= 6#FgoF?F2F?" }{TEXT -1 18 " or\n\n\n\n " }{XPPEDIT 808 1 "w[ 3] = MATRIX( [ [0],[0],[2],[1] ] ) + MATRIX( [ [1/sqrt (3)] , [0],[-1/sqrt(3)] , [1/sqrt(3)] ] ) /sqrt(3) - 4* MAT RIX( [ [-1/sqrt(15)],[3/sqrt(15)],[1/sqrt(15)],[2/sqrt(15)] ] ) /sqrt(15)" "/&%\"wG6#\"\"$,(-%'MATRIXG6#7&7#\"\"!7#F-7#\"\"#7#\"\"\"F2 *&-F)6#7&7#*&F2F2-%%sqrtG6#F&!\"\"7#F-7#,$*&F2F2-F:6#F&F = (9 + 16+49+1 6)/25" "/-%-anglebracketG6$&%\"wG6#\"\"$&F'6#F)*&,*\"\"*\"\"\"\"#;F/\" #\\F/F0F/F/\"#D!\"\"" }{TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 5 " so " }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{XPPEDIT 811 1 " sqrt( ) = 3*sqrt(10)/5" "/-%%sqrtG6#-%-anglebracketG6$&% \"wG6#\"\"$&F*6#F,*(F,\"\"\"-F$6#\"#5F0\"\"&!\"\"" }{TEXT -1 4 " ." }}{PARA 0 "" 0 "" {TEXT -1 3 "\n " }}{PARA 0 "" 0 "" {TEXT -1 14 "The refore " }{XPPEDIT 812 1 "v[3] = 5*MATRIX( [ [3/5],[-4/5],[7/5 ],[4/5] ] ) /(3*sqrt(10))" "/&%\"vG6#\"\"$*(\"\"&\"\"\"-%'MATRIX G6#7&7#*&F&F)F(!\"\"7#,$*&\"\"%F)F(F0F07#*&\"\"(F)F(F07#*&F4F)F(F0F)*& F&F)-%%sqrtG6#\"#5F)F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 813 1 "v[3] = MATRIX( [ [1/sqrt(10)],[-4/3/sqrt(10)],[7/3/sqrt(10 )],[4/3/sqrt(10)] ] ) " "/&%\"vG6#\"\"$-%'MATRIXG6#7&7#*&\"\"\"F- -%%sqrtG6#\"#5!\"\"7#,$*(\"\"%F-F&F2-F/6#F1F2F27#*(\"\"(F-F&F2-F/6#F1F 27#*(F6F-F&F2-F/6#F1F2" }{TEXT 814 1 "." }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 14 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "As a che ck" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "v[1] := vector([ 1/sqrt(3) \+ , 0 , -1/sqrt(3) , 1/sqrt(3) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"vG6#\"\"\"-%'VECTORG6#7&,$*$\"\"$#F'\"\"##F'F.\"\"!,$F-#!\"\"F.F ," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "v[2] := vector([-1/sqr t(15), 3/sqrt(15),1/sqrt(15), 2/sqrt(15)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"#-%'VECTORG6#7&,$*$\"#:#\"\"\"F'#!\"\"F.,$ F-#F0\"\"&,$F-#F0F.,$F-#F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "v[3] := vector([ 1/sqrt(10) , -4/3/sqrt(10) , 7/3/sqrt(10) , \+ 4/3/sqrt(10) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"$-%'V ECTORG6#7&,$*$\"#5#\"\"\"\"\"##F0F.,$F-#!\"#\"#:,$F-#\"\"(\"#I,$F-#F1F 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "for i from 1 to 3 do \n for j from 1 to 3 do\n print( 'dotprod'(v[i],v[j]) = dotprod (v[i],v[j]));\n od;\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotp rodG6$&%\"vG6#\"\"\"F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotpro dG6$&%\"vG6#\"\"\"&F(6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%(dotprodG6$&%\"vG6#\"\"\"&F(6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotprodG6$&%\"vG6#\"\"#&F(6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotprodG6$&%\"vG6#\"\"#F'\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotprodG6$&%\"vG6#\"\"#&F(6#\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotprodG6$&%\"vG6#\"\"$&F (6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotprodG6$&%\"vG 6#\"\"$&F(6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%(dotprodG 6$&%\"vG6#\"\"$F'\"\"\"" }}}{PARA 0 "" 0 "" {TEXT 807 93 "____________ ______________________________________________________________________ ___________" }}{PARA 0 "" 0 "" {TEXT 772 6 "Maple:" }{TEXT -1 9 " U se " }{TEXT 781 5 "Maple" }{TEXT -1 49 " to do the calculations of t he preceding example" }{TEXT 778 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 773 9 "Solution:" }{TEXT -1 4 " \n" }}{PARA 0 "" 0 "" {TEXT -1 50 "There is a command to do the \+ Gram-Schmidt process." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; \+ with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition \+ for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for t race" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "u[1] := vector([ 1 \+ , 0 , -1 , 1 ] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"\"-% 'VECTORG6#7&F'\"\"!!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "u[2] := vector([ 0 , 1 , 0 , 1 ] ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"#-%'VECTORG6#7&\"\"!\"\"\"F,F-" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "u[3] := vector( [ 0 , 0 , 2 , 1 ] );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"$-%'VECTORG6#7&\"\"!F,\" \"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "GramSchmidt( [ u[1], u[2], u[3] ] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7&\"\"\"\" \"!!\"\"F%7&#F'\"\"$F%#F%F*#\"\"#F*7&#F*\"\"&#!\"%F0#\"\"(F0#\"\"%F0" }}}{PARA 0 "" 0 "" {TEXT 818 93 "_____________________________________ ________________________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 819 11 "Definition:" }{TEXT -1 11 " If " }{XPPEDIT 831 1 "U" "I\"UG6\"" }{TEXT -1 44 " is a subspace of an inner product sp ace " }{XPPEDIT 830 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$ %$~.~GF'" }{TEXT -1 25 " then we define\n\n " }{XPPEDIT 832 1 " U^`~` = \{v*` in `*V*`| `* = 0*` for all `*u*` in `*U" "/)%\"UG %\"|irG<#/*,%\"vG\"\"\"%%~in~GF*%\"VGF*%#|gr~GF*-%-anglebracketG6$%\"u GF)F**,\"\"!F*%,~~~for~all~GF*F1F*F+F*F$F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "For reasons tha t will be apparent when we prove the next theorem, " }{XPPEDIT 835 1 "U^`~` " ")%\"UG%\"|irG" }{TEXT -1 16 " is called the " }{TEXT 836 21 "orthogonal complement" }{TEXT -1 7 " of " }{XPPEDIT 834 1 "U" " I\"UG6\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 833 8 "Theorem:" }{TEXT -1 7 " \nIf " }{XPPEDIT 838 1 "U" "I\"UG6\"" }{TEXT -1 43 " is a subsp ace of an inner product space " }{XPPEDIT 837 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$%$~.~GF'" }{TEXT -1 35 ", then its orthogon al complement " }{XPPEDIT 839 1 "U^`~` " ")%\"UG%\"|irG" }{TEXT -1 25 " is also a subspace of " }{XPPEDIT 840 1 "V" "I\"VG6\"" }{TEXT -1 18 ". Furthermore, " }{XPPEDIT 841 1 "V" "I\"VG6\"" }{TEXT -1 23 " is the direct sum of " }{XPPEDIT 842 1 "U" "I\"UG6\"" }{TEXT -1 8 " and " }{XPPEDIT 843 1 "U^`~` " ")%\"UG%\"|irG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 844 6 "Proof: " }{TEXT -1 27 " The verification that " }{XPPEDIT 845 1 "U^`~` " ")%\"UG%\"|irG" }{TEXT -1 95 " is closed under addition and scalar mu ltiplication is routine. We must show that any vector " }{XPPEDIT 846 1 "v" "I\"vG6\"" }{TEXT -1 8 " in " }{XPPEDIT 847 1 "V" "I\"VG 6\"" }{TEXT -1 31 " has a unique decomposition as" }}{PARA 0 "" 0 "" {TEXT -1 33 " \n " }{XPPEDIT 848 1 "v=u+ u^`~`" "/%\"vG,&%\"uG\"\"\")F%%\"|irGF&" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "with " }{XPPEDIT 850 1 "u" "I\"uG6\"" }{TEXT -1 6 " in " }{XPPEDIT 849 1 "U" "I\"UG6 \"" }{TEXT -1 7 " and " }{XPPEDIT 852 1 "u^`~` " ")%\"uG%\"|irG" } {TEXT -1 7 " in " }{XPPEDIT 851 1 "U^`~` " ")%\"UG%\"|irG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{XPPEDIT 854 1 "\{ u[1],u[2 ],`...`,u[k] \}" "<&&%\"uG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"kG" }{TEXT -1 31 " be an orthonormal basis of " }{XPPEDIT 855 1 "U" "I\"UG6\" " }{TEXT -1 74 " . Such a basis exists by the Gram-Schmidt orthogonal ization process. Let" }}{PARA 0 "" 0 "" {TEXT -1 20 " \n \+ " }{XPPEDIT 853 1 "u = *u[1]+*u[2]+`...`+* u[k]" "/%\"uG,**&-%-anglebracketG6$%\"vG&F#6#\"\"\"F,&F#6#F,F,F,*&-F'6 $F)&F#6#\"\"#F,&F#6#F4F,F,%$...GF,*&-F'6$F)&F#6#%\"kGF,&F#6#F=F,F," } {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 " " {TEXT -1 16 " " }{XPPEDIT 858 1 "u^`~` =v-u" "/)%\"uG %\"|irG,&%\"vG\"\"\"F$!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Clearly " }{XPPEDIT 860 1 "u " "I\"uG6\"" }{TEXT -1 14 " belongs to " }{XPPEDIT 859 1 "U" "I\"UG6 \"" }{TEXT -1 26 ". Also, by orthogonality," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{XPPEDIT 861 1 " = - *" "/-%-anglebracketG6$) %\"uG%\"|irG&F'6#%\"iG,&-F$6$%\"vG&F'6#F+\"\"\"*&-F$6$F/&F'6#F+F2-F$6$ &F'6#F+&F'6#F+F2!\"\"" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "\nwhich is zero. It follows that " }{XPPEDIT 862 1 "u^`~` " ")% \"uG%\"|irG" }{TEXT -1 35 " is orthogonal to all vectors in " } {XPPEDIT 863 1 "U" "I\"UG6\"" }{TEXT -1 29 " and therefore belongs to " }{XPPEDIT 864 1 "U^`~`" ")%\"UG%\"|irG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The decompositi on " }{XPPEDIT 865 1 "v=u+u^`~`" "/%\"vG,&%\"uG\"\"\")F%%\"|irG F&" }{TEXT -1 19 " is unique. If " }{XPPEDIT 866 1 "v=u[0]+u[0]^`~ `" "/%\"vG,&&%\"uG6#\"\"!\"\"\")&F&6#F(%\"|irGF)" }{TEXT -1 35 " wer e another, then we would have" }}{PARA 0 "" 0 "" {TEXT -1 13 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 24 " " } {XPPEDIT 868 1 "u-u[0] = u[0]^`~`-u^`~`" "/,&%\"uG\"\"\"&F$6#\"\"!!\" \",&)&F$6#F(%\"|irGF%)F$F.F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The right side, being an element of " }{XPPEDIT 869 1 "U^`~` " ")%\"UG%\"|irG" }{TEXT -1 64 ", would have to be orthogonal to the \+ left side, an element of " }{XPPEDIT 870 1 "U" "I\"UG6\"" }{TEXT -1 33 " . In other words, the element " }{XPPEDIT 871 1 "u-u[0] " ",&% \"uG\"\"\"&F#6#\"\"!!\"\"" }{TEXT -1 56 " would have 0 inner product \+ and therefore be 0. Thus, " }{XPPEDIT 873 1 "u = u[0] " "/%\"uG&F#6# \"\"!" }{TEXT -1 29 ". It would then follow that " }{XPPEDIT 872 1 "u [0]^`~` = u^`~`" "/)&%\"uG6#\"\"!%\"|irG)F%F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 874 11 "Definition:" }{TEXT -1 20 " Suppose that \+ " }{XPPEDIT 875 1 "[V,<` . ` , ` . `>" "7$%\"VG-%-anglebracketG6$%$~. ~GF'" }{TEXT -1 50 " is an inner product space. Suppose also that \+ " }{XPPEDIT 881 1 "T" "I\"TG6\"" }{TEXT -1 26 " is a linear map from \+ " }{XPPEDIT 882 1 "V" "I\"VG6\"" }{TEXT -1 4 " to " }{XPPEDIT 883 1 "V" "I\"VG6\"" }{TEXT -1 9 ". If " }{XPPEDIT 884 1 "T^`*`" ")%\" TG%\"*G" }{TEXT -1 30 " is also a linear map from " }{XPPEDIT 885 1 "V" "I\"VG6\"" }{TEXT -1 4 " to " }{XPPEDIT 886 1 "V" "I\"VG6\"" } {TEXT -1 20 " and if \n\n " }{XPPEDIT 877 1 " = " "/-%-anglebracketG6$-%\"TG6#%\"uG%\"vG-F$6$F)-)F'%\"*G6#F* " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "for all " }{XPPEDIT 892 1 "u" "I\"uG6\"" }{TEXT -1 6 " an d " }{XPPEDIT 891 1 "v" "I\"vG6\"" }{TEXT -1 7 " in " }{XPPEDIT 890 1 "V" "I\"VG6\"" }{TEXT -1 20 " then we say that " }{XPPEDIT 889 1 "T^`*`" ")%\"TG%\"*G" }{TEXT -1 9 " is the " }{TEXT 888 8 " adj oint" }{TEXT -1 7 " of " }{XPPEDIT 887 1 "T" "I\"TG6\"" }{TEXT -1 69 ". \n\n\n\nThe property of being an adjoint is a symmetric relati on: " }{XPPEDIT 895 1 "T^`*`" ")%\"TG%\"*G" }{TEXT -1 20 " is the a djoint of " }{XPPEDIT 893 1 "T" "I\"TG6\"" }{TEXT -1 18 " if and onl y if " }{XPPEDIT 898 1 "T " "I\"TG6\"" }{TEXT -1 21 " is the adjoint \+ of " }{XPPEDIT 896 1 "T^`*`" ")%\"TG%\"*G" }{TEXT 899 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 900 8 "Theorem :" }{TEXT -1 20 " Suppose that "