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

%%EndSetup
%%Page: 1 1
1 0 bop 162 274 a Fh(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(9)162 394 y(Prof.)g(Dr.)h(B.)f(P)n(areigis)293 644
y Fg(Lineare)23 b(Algebra)g(und)h(analytisc)n(he)f(Geometrie)g(I)r(I)
222 746 y Ff(1.)h(Seien)17 b Fe(m)g Ff(und)h Fe(n)g Ff(ganze)f(Zahlen)h
(und)g Fe(g)h Ff(der)f(gr)1187 748 y(\177)1187 746 y(o\031te)g
(gemeinsame)d(T)l(eiler)h(v)o(on)284 806 y Fe(m)h Ff(und)h
Fe(n)p Ff(.)f(Wir)g(k)629 808 y(\177)629 806 y(onnen)h(also)g(ganze)f
(Zahlen)h Fe(m)1214 788 y Fd(0)1242 806 y Ff(und)g Fe(n)1370
788 y Fd(0)1399 806 y Ff(mit)e Fe(m)g Ff(=)g Fe(g)r(m)1671
788 y Fd(0)1700 806 y Ff(und)284 867 y Fe(n)e Ff(=)g
Fe(g)r(n)433 849 y Fd(0)461 867 y Ff(\014nden.)i(Zeigen)f(Sie,)g
(da\031)646 967 y Fc(Z)682 974 y Fb(g)713 967 y Fa(!)f
Ff(Hom)n(\()p Fc(Z)934 974 y Fb(m)964 967 y Fe(;)8 b
Fc(Z)1022 974 y Fb(n)1042 967 y Ff(\))p Fe(;)1101 954
y Ff(\026)1100 967 y Fe(k)15 b Fa(7!)f Ff(\()1223 954
y(\026)1223 967 y Fe(l)g Fa(7!)p 1316 926 84 2 v 14 w
Fe(k)r(l)q(n)1388 952 y Fd(0)1400 967 y Ff(\))284 1067
y(ein)h(Grupp)q(enisomorphism)o(us)g(ist.)284 1146 y(\(Hin)o(w)o(eis:)
504 1136 y(\177)498 1146 y(Ub)q(erlegen)h(Sie)g(sic)o(h,)g(da\031)i
Fe(n)1041 1128 y Fd(0)1070 1146 y Ff(und)g Fe(m)1212
1128 y Fd(0)1240 1146 y Ff(teilerfremd)c(sind,)j(und)g(da\031)284
1206 y(daraus)j(folgt,)f(da\031)h(sic)o(h,)d(wie)i(in)f(Aufgab)q(e)i
(40)f(auf)h(Blatt)e(10)i(im)d(WS)i(00/01,)284 1267 y(1)d(durc)o(h)g
Fe(n)490 1249 y Fd(0)518 1267 y Ff(und)h Fe(m)659 1249
y Fd(0)686 1267 y Ff(ausdr)803 1269 y(\177)802 1267 y(uc)o(k)o(en)f(l)
954 1269 y(\177)954 1267 y(a\031t.\))664 b(\(5\))222
1365 y(2.)24 b(Sei)15 b Fe(f)20 b Ff(ein)15 b(Endomorphism)o(us)f(des)h
(endlic)o(hdimensionalen)d(V)l(ektorraumes)i Fe(V)d Ff(.)284
1426 y(Dann)22 b(k)m(ann)h Fe(V)33 b Ff(mit)20 b(Hilfe)g(v)o(on)h
Fe(f)27 b Ff(zu)22 b(einem)d(Mo)q(dul)1349 1428 y(\177)1348
1426 y(ub)q(er)j(dem)e(P)o(olynom-)284 1486 y(ring)14
b Fe(K)t Ff([)p Fe(x)p Ff(])f(gemac)o(h)o(t)f(w)o(erden.)h(Zeigen)h
(Sie,)f(da\031)i Fe(f)k Ff(genau)c(dann)g(diagonalisier-)284
1546 y(bar)i(ist,)g(w)o(enn)g Fe(V)28 b Ff(als)18 b Fe(K)t
Ff([)p Fe(x)p Ff(]-Mo)q(dul)e(direkte)g(Summe)e(v)o(on)j(Un)o(termo)q
(duln)f(ist,)284 1606 y(die)f(gleic)o(hzeitig)f(eindimensionale)g(Un)o
(terr)1108 1608 y(\177)1108 1606 y(aume)g(sind.)380 b(\(5\))222
1705 y(3.)24 b(Seien)18 b Fe(f)25 b Ff(und)19 b Fe(g)j
Ff(zw)o(ei)c(diagonalisierbare)h(Endomorphismen)e(des)j(endlic)o(hdi-)
284 1765 y(mensionalen)13 b(V)l(ektorraumes)g Fe(V)e
Ff(.)k(Zeigen)f(Sie,)f(da\031)j(folgende)e(Aussagen)1674
1767 y(\177)1674 1765 y(aqui-)284 1825 y(v)m(alen)o(t)h(sind:)304
1924 y(\(a\))25 b Fe(f)d Ff(und)16 b Fe(g)j Ff(k)o(omm)n(utiere)o(n,)13
b(d.)j(h.)g Fe(f)g Fa(\016)11 b Fe(g)16 b Ff(=)e Fe(g)f
Fa(\016)e Fe(f)5 b Ff(.)302 2002 y(\(b\))24 b Fe(f)e
Ff(und)17 b Fe(g)h Ff(sind)f(gleic)o(hzeitig)d(diagonalisierbar,)i(d.)g
(h.)g(es)h(gibt)f(eine)g(Basis)391 2063 y(v)o(on)g Fe(V)c
Ff(,)j(die)h(aus)h(Eigen)o(v)o(ektoren)e(v)o(on)h(so)o(w)o(ohl)g
Fe(f)22 b Ff(als)16 b(auc)o(h)g Fe(g)j Ff(b)q(esteh)o(t.)1718
2161 y(\(5\))222 2260 y(4.)24 b(Sei)10 b Fe(V)23 b Ff(ein)11
b(endlic)o(hdim)o(ensionaler)d(unit)1023 2262 y(\177)1023
2260 y(arer)j(V)l(ektorraum.)f(Ein)h(Endomorphis-)284
2320 y(m)o(us)20 b(v)o(on)i Fe(V)32 b Ff(hei\031t)22
b(normal,)e(falls)h(er)g(mit)f(seiner)h(adjungierten)g(Abbildung)284
2381 y(k)o(omm)o(uti)o(ert,)14 b(d.)j(h.)h(falls)f Fe(f)g
Fa(\016)12 b Fe(f)890 2363 y Fb(ad)946 2381 y Ff(=)k
Fe(f)1029 2363 y Fb(ad)1080 2381 y Fa(\016)c Fe(f)23
b Ff(gilt.)17 b(Zeigen)g(Sie,)g(da\031)h Fe(f)23 b Ff(genau)284
2441 y(dann)h(normal)f(ist,)g(w)o(enn)h(es)f(eine)g(Orthonormalbasis)h
(aus)g(Eigen)o(v)o(ektoren)284 2501 y(v)o(on)16 b Fe(f)22
b Ff(gibt.)p eop
%%Page: 2 2
2 1 bop 284 274 a Ff(\(Hin)o(w)o(eis:)17 b(Zeigen)h(Sie)g(zun)814
276 y(\177)814 274 y(ac)o(hst,)g(da\031)i Fe(f)25 b Ff(und)19
b Fe(f)1229 256 y Fb(ad)1287 274 y Ff(denselb)q(en)g(Kern)g(hab)q(en,)
284 334 y(und)387 336 y(\177)386 334 y(ub)q(erlegen)h(Sie)h(dann,)g(w)o
(as)g(das)h(f)1041 336 y(\177)1040 334 y(ur)f(die)f(Eigenr)1329
336 y(\177)1329 334 y(aume)f(v)o(on)i Fe(f)26 b Ff(und)21
b Fe(f)1741 316 y Fb(ad)284 394 y Ff(b)q(edeutet.\))1215
b(\(5\))162 509 y(Abgab)q(e:)17 b(Mitt)o(w)o(o)q(c)o(h,)f(11.7.2001,)i
(9.00)g(Uhr,)1039 498 y(\177)1033 509 y(Ubungsk)1220
511 y(\177)1220 509 y(asten)g(im)d(1.)i(Sto)q(c)o(k)g(des)g(Ma-)162
569 y(thematisc)o(hen)d(Instituts.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF