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%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300dpi
TeXDict begin
%%PaperSize: a4

%%EndSetup
%%Page: 1 1
1 0 bop 162 274 a Fi(Mathematisc)n(hes)17 b(Institut)826
b(SS)19 b(2001)162 334 y(der)f(Univ)n(ersit)487 336 y(\177)487
334 y(at)g(M)609 336 y(\177)607 334 y(unc)n(hen)828 b(Blatt)18
b(7)162 394 y(Prof.)g(Dr.)h(B.)f(P)n(areigis)293 691
y Fh(Lineare)23 b(Algebra)g(und)h(analytisc)n(he)f(Geometrie)g(I)r(I)
222 805 y Fg(1.)h(Sei)17 b Ff(A)g Fg(eine)g(reelle)f(orthogonale)j(2)13
b Fe(\002)f Fg(2-Matrix.)17 b(Zeigen)g(Sie,)g(da\031)i(die)e(Deter-)284
865 y(minan)o(te)d(v)o(on)i Ff(A)g Fg(en)o(t)o(w)o(eder)f(1)h(o)q(der)h
Fe(\000)p Fg(1)f(ist.)g(Zeigen)f(Sie)h(w)o(eiter:)304
967 y(\(a\))25 b(Ist)17 b(die)f(Determinan)o(te)f(v)o(on)h
Ff(A)h Fg(gleic)o(h)f(1,)g(so)i(gibt)f(es)g(eine)f(reelle)f(Zahl)h
Ff(\013)391 1027 y Fg(mit)822 1104 y Ff(A)d Fg(=)924
1034 y Fd(\022)961 1073 y Fg(cos\()p Ff(\013)p Fg(\))42
b Fe(\000)8 b Fg(sin\()p Ff(\013)p Fg(\))964 1133 y(sin\()p
Ff(\013)p Fg(\))65 b(cos\()p Ff(\013)p Fg(\))1313 1034
y Fd(\023)302 1230 y Fg(\(b\))24 b(Ist)14 b(die)f(Determinan)o(te)f(v)o
(on)i Ff(A)f Fg(gleic)o(h)g Fe(\000)p Fg(1,)h(so)g(gibt)h(es)e(eine)h
(reelle)e(Zahl)i Ff(\014)391 1291 y Fg(mit)820 1367 y
Ff(A)g Fg(=)923 1297 y Fd(\022)959 1337 y Fg(cos)q(\()p
Ff(\014)s Fg(\))67 b(sin)o(\()p Ff(\014)s Fg(\))962 1397
y(sin\()p Ff(\014)s Fg(\))43 b Fe(\000)8 b Fg(cos)q(\()p
Ff(\014)s Fg(\))1315 1297 y Fd(\023)284 1498 y Fg(\(Hin)o(w)o(eis:)16
b(Wie)i(in)g(der)g(V)l(orlesung)g(de\014niert)f(wurde,)h(hei\031t)h
(eine)e(reelle)f(qua-)284 1558 y(dratisc)o(he)f(Matrix)h
Ff(A)g Fg(orthogonal,)h(w)o(enn)f Ff(A)1127 1540 y Fc(T)1154
1558 y Ff(A)e Fg(=)f Ff(E)20 b Fg(gilt.\))302 b(\(6\))222
1660 y(2.)24 b(Sei)17 b Ff(V)30 b Fg(ein)17 b(V)l(ektorraum)775
1662 y(\177)774 1660 y(ub)q(er)h(dem)f(K)1034 1662 y(\177)1034
1660 y(orp)q(er)h Ff(K)t Fg(.)g(Ist)g Ff(f)k Fg(:)17
b Ff(V)28 b Fe(!)17 b Ff(V)29 b Fg(ein)18 b(Endo-)284
1720 y(morphism)o(us)c(und)761 1816 y Ff(p)p Fg(\()p
Ff(X)t Fg(\))h(=)959 1753 y Fc(n)933 1768 y Fd(X)937
1874 y Fc(k)q Fb(=0)1014 1816 y Ff(a)1040 1823 y Fc(k)1061
1816 y Ff(X)1105 1795 y Fc(k)1140 1816 y Fe(2)f Ff(K)t
Fg([)p Ff(X)t Fg(])284 1945 y(ein)19 b(P)o(olynom)g(mit)g(Ko)q
(e\016zien)o(ten)g(aus)i Ff(K)t Fg(,)e(so)i(k)m(ann)g(man)f
Ff(f)25 b Fg(in)20 b Ff(p)h Fg(einsetzen)284 2005 y(und)16
b(erh)449 2007 y(\177)449 2005 y(alt)g(dann)h(die)f(lineare)f
(Abbildung)865 2150 y Ff(p)p Fg(\()p Ff(f)5 b Fg(\))14
b(=)1047 2088 y Fc(n)1022 2103 y Fd(X)1026 2209 y Fc(k)q
Fb(=0)1102 2150 y Ff(a)1128 2157 y Fc(k)1149 2150 y Ff(f)1178
2130 y Fc(k)284 2306 y Fg(Zeigen)d(Sie:)f(Ist)i Ff(\025)g
Fg(ein)f(Eigen)o(w)o(ert)g(v)o(on)g Ff(f)5 b Fg(,)12
b(so)g(ist)g Ff(p)p Fg(\()p Ff(\025)p Fg(\))g(ein)f(Eigen)o(w)o(ert)g
(v)o(on)g Ff(p)p Fg(\()p Ff(f)5 b Fg(\).)1718 2386 y(\(4\))p
eop
%%Page: 2 2
2 1 bop 222 274 a Fg(3.)24 b(Der)16 b(Einheitsw)589 276
y(\177)588 274 y(urfel)f(im)f Fa(R)809 256 y Fc(n)846
274 y Fg(ist)i(die)g(Menge)583 384 y Ff(W)629 391 y Fc(n)666
384 y Fg(:=)e Fe(f)p Fg(\()p Ff(x)804 391 y Fb(1)823
384 y Ff(;)8 b(:)g(:)g(:)16 b(;)8 b(x)969 391 y Fc(n)992
384 y Fg(\))14 b Fe(j)f(8)p Ff(i)f Fe(\024)i Ff(n)g Fg(:)f(0)i
Fe(\024)e Ff(x)1352 391 y Fc(i)1380 384 y Fe(\024)g Fg(1)p
Fe(g)284 494 y Fg(Erkl)376 496 y(\177)376 494 y(aren)e(Sie,)f(w)o(elc)o
(her)g(V)l(ektor)h(die)g(Diagonale)h(in)f(diesem)e(W)1443
496 y(\177)1442 494 y(urfel)i(b)q(esc)o(hreibt,)284 554
y(und)j(b)q(estimmen)c(Sie)j(den)g(Kosin)o(us)h(des)g(Wink)o(els)e
(zwisc)o(hen)g(dieser)h(Diagonale)284 614 y(und)j(der)g(Seite,)e(die)i
(durc)o(h)f(den)h(ersten)g(k)m(anonisc)o(hen)g(Basisv)o(ektor)f(b)q
(estimm)o(t)284 675 y(ist.)g(Geb)q(en)i(Sie)e(diesen)h(Wink)o(el)f(f)
927 677 y(\177)926 675 y(ur)h Ff(n)e Fg(=)g(2)i(und)h
Ff(n)d Fg(=)g(4)i(explizit)e(an.)125 b(\(4\))222 776
y(4.)24 b(Bestimm)o(en)15 b(Sie)i(un)o(ter)h(V)l(erw)o(endung)g(eines)f
(Computeralgebraprogrammes)284 836 y(\(wie)c(et)o(w)o(a)g(Maple)f(o)q
(der)i(Mathematica\))e(dasjenige)i(reelle)d(P)o(olynom)h
Ff(p)i Fg(vierten)284 897 y(Grades,)i(das)h(die)f(Bedingung)823
1047 y Ff(p)p Fg(\()p Ff(x)p Fg(\))e(=)979 962 y Fd(\()1019
1013 y Fg(0)49 b(:)13 b Ff(x)h Fe(6)p Fg(=)g(0)1019 1086
y(1)49 b(:)13 b Ff(x)h Fg(=)g(0)284 1198 y(in)i(den)g(Punkten)g
Ff(x)e Fg(=)f Fe(\000)p Fg(3)p Ff(;)8 b Fe(\000)p Fg(2)p
Ff(;)g Fe(\000)p Fg(1)p Ff(;)g Fg(0)p Ff(;)g Fg(1)p Ff(;)g
Fg(2)p Ff(;)g Fg(3)18 b(am)d(b)q(esten)h(appro)o(ximiert.)284
1279 y(\(Hin)o(w)o(eis:)i(Geb)q(en)j(Sie)f(bitte)g(einen)g(Ausdruc)o(k)
g(des)h(mit)d(Hilfe)h(des)i(Compu-)284 1339 y(teralgebraprogrammes)16
b(erstellten)f(Arb)q(eitsblattes)i(ab,)g(auf)h(dem)d(Sie)i(hand-)284
1400 y(sc)o(hriftlic)o(h)i(detailliert)g(erl)795 1402
y(\177)795 1400 y(autern,)h(w)o(as)i(in)f(den)g(einzelnen)e(Rec)o
(hensc)o(hritten)284 1460 y(gesc)o(hieh)o(t.)14 b(Lassen)j(Sie)f(auf)g
(dem)f(Blatt)g(auc)o(h)h(einen)g(Plot)g(des)g(P)o(olynoms)f(an-)284
1520 y(fertigen.\))1239 b(\(6\))162 1634 y(Abgab)q(e:)17
b(Mitt)o(w)o(o)q(c)o(h,)f(27.6.2001,)i(9.00)g(Uhr,)1039
1624 y(\177)1033 1634 y(Ubungsk)1220 1636 y(\177)1220
1634 y(asten)g(im)d(1.)i(Sto)q(c)o(k)g(des)g(Ma-)162
1694 y(thematisc)o(hen)d(Instituts.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF