{VERSION 4 0 "HP RISC UNIX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 51 "Beispiel einer Orthogona lisierung nach Gram-Schmidt" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg):" }}}{EXCHG {PARA 257 " " 0 "" {TEXT -1 59 "Gesucht ist eine Orthonormalbasis des Untervektorr aums von " }{TEXT 257 0 "" }{XPPEDIT 18 0 "C^3;" "6#*$%\"CG\"\"$" } {TEXT 256 0 "" }{TEXT -1 54 ", der von den folgenden drei Vektoren auf gespannt ist:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "b1 := vec tor([1, I, -I, -1]);\nb2 := vector([1+2*I, -2+I, -I, -1]);\nb3 := vect or([1+2*I, -I, -4-I, 1+2*I]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 30 "Zun\344chst suchen wir eine Ortho" }{TEXT 258 5 "gonal" }{TEXT -1 8 "basis:\n\n" }{TEXT 260 15 "Erster Schritt:" }{TEXT -1 34 " L\366se \+ das Teilproblem f\374r den von " }{XPPEDIT 18 0 "b[1];" "6#&%\"bG6#\" \"\"" }{TEXT -1 30 " aufgespannten Untervektoraum." }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c1 := b1;" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 261 16 "Zweiter Schritt:" }{TEXT -1 34 " L\366 se das Teilproblem f\374r den von " }{XPPEDIT 18 0 "b[1],b[2];" "6$&% \"bG6#\"\"\"&F$6#\"\"#" }{TEXT -1 38 " aufgespannten Untervektoraum.\n Ansatz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "c2 := b2 + lambd a*c1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(c1, c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(b2, c1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(dotprod(c2, c1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "lambda := solve(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "c2 := evalm(b2 + lambda*c1); " }}}{EXCHG {PARA 261 "" 0 "" {TEXT 262 16 "Dritter Schritt:" }{TEXT 258 1 " " }{TEXT -1 17 "L\366se das Teilprob" }{TEXT 259 0 "" }{TEXT -1 16 "lem f\374r den von " }{XPPEDIT 18 0 "b[1],b[2],b[3];" "6%&%\"bG 6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT 256 1 " " }{TEXT -1 37 "aufgespan nten Untervektoraum.\nAnsatz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "c3 := b3 + mu*c1 + nu*c2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(b3, c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(b3, c2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simpli fy(dotprod(c3, c1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "mu \+ := solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(do tprod(c3, c2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nu := so lve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "c3 := evalm(b3 + mu*c1 + nu*c2);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 106 "Damit ist \+ eine Orthogonalbasis gefunden; ihre Elemente m\374ssen nun nur noch au f L\344nge eins normiert werden:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(c1, c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(c2, c2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dotprod(c3, c3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e1 := 1/2*evalm(c1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e2 := 1/2*evalm(c2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "e3 := 1/sqrt(8)*evalm(c3);" }}}}{MARK "1 0 0" 22 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }