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003890C7FCEA7C0FEAFE0E131E131C5BEA74F0EA3FE0EA0F8020277F9A23>121
D E /Fh 29 128 df<1238127C12FEA3127C123807077C8610>46
D<13FE3807FFC0380F83E0381F01F0383E00F8A248137CA312FC147EAD007C137CA36C13
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15>73 D<D8FFF0EC0FFF6D5C000716E0D806FC1437A3017E1467A26D14C7A290391F8001
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99 D<EB01FEA2EB003EABEA01FC3807FFBE380F81FE381F007E003E133E127E127C12FC
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108 D<3AFF07F007F090391FFC1FFC3A1F303E303E01401340496C487EA201001300AE3B
FFE0FFE0FFE0A22B167E9530>I<38FF07E0EB1FF8381F307CEB403CEB803EA21300AE39
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1FEB3FC0381E67E013C71387A2381F83C090C7FCADEAFFF0A213167E9517>114
D<EA0FF3EA3FFFEA781FEA6007EAE003A212F000FCC7FCEA7FE013F8EA3FFEEA0FFF1201
38000F80EAC007130312E0A238F00700EAFC0EEAEFFCEAC7F011167E9516>I<487EA412
03A21207A2120F123FB5FCA2EA0F80ABEB8180A5EB8300EA07C3EA03FEEA00F811207F9F
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00EA380E11077BA31C>127 D E end
%%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(2)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)
222 751 y Ff(1.)h(Seien)19 b Fe(U)26 b Ff(und)21 b Fe(W)27
b Ff(Un)o(terr)791 753 y(\177)791 751 y(aume)19 b(des)h(V)l
(ektorraumes)f Fe(V)11 b Ff(.)20 b(Es)h(gelte)f Fe(W)28
b Fd(\032)21 b Fe(U)5 b Ff(.)284 812 y(Zeigen)11 b(Sie,)h(da\031)h
Fe(U)r(=W)20 b Ff(ein)12 b(Un)o(terraum)e(des)j(F)l(aktorraumes)e
Fe(V)5 b(=W)21 b Ff(ist.)12 b(Zeigen)284 872 y(Sie)j(den)i(2.)f
(Isomorphiesatz)789 976 y(\()p Fe(V)5 b(=W)i Ff(\))p
Fe(=)p Ff(\()p Fe(U)r(=W)g Ff(\))1127 963 y Fd(\030)1127
978 y Ff(=)1179 976 y Fe(V)e(=U)284 1081 y Ff(indem)19
b(Sie)h(den)h(Homomorphiesatz)d(auf)k(eine)e(k)m(anonisc)o(he)g
(Abbildung)h(v)o(on)284 1141 y Fe(V)5 b(=W)24 b Ff(nac)o(h)16
b Fe(V)5 b(=U)23 b Ff(an)o(w)o(enden.)856 b(\(6\))222
1241 y(2.)24 b(Sei)17 b(\()p Fe(V)s(;)8 b(A)p Ff(\))18
b(ein)f(a\016ner)h(Raum.)f Fe(GL)p Ff(\()p Fe(V)12 b
Ff(\))18 b(sei)g(die)f(Grupp)q(e)i(der)f(Automorphis-)284
1302 y(men)e(v)o(on)h Fe(V)11 b Ff(.)17 b(Ist)g Fe(p)651
1309 y Fc(0)687 1302 y Fd(2)f Fe(A)h Ff(fest)g(gegeb)q(en,)g(so)h(sei)f
(f)1231 1304 y(\177)1230 1302 y(ur)g Fe(v)g Fd(2)f Fe(V)28
b Ff(und)18 b Fe(f)j Fd(2)16 b Fe(GL)p Ff(\()p Fe(V)c
Ff(\))284 1362 y(die)j(Abbildung)h Fe(A)639 1369 y Fb(v)q(;f)706
1362 y Ff(de\014niert)f(durc)o(h)638 1466 y Fe(A)675
1473 y Fb(v)q(;f)739 1466 y Ff(:)e Fe(A)h Fd(!)f Fe(A;)8
b(p)14 b Fd(7!)g Fe(v)e Ff(+)f Fe(f)5 b Ff(\()p Fe(p)12
b Fd(\000)f Fe(p)1284 1473 y Fc(0)1304 1466 y Ff(\))g(+)g
Fe(p)1407 1473 y Fc(0)304 1571 y Ff(\(a\))25 b(Zeigen)12
b(Sie,)g(da\031)i(jede)e(A\016nit)956 1573 y(\177)956
1571 y(at)g(v)o(on)g(der)h(F)l(orm)f Fe(A)1342 1578 y
Fb(v)q(;f)1405 1571 y Ff(ist,)g(und)h(da\031)h(dab)q(ei)391
1631 y Fe(v)k Ff(und)e Fe(f)22 b Ff(eindeutig)15 b(b)q(estimm)o(t)e
(sind.)j(Zeigen)g(Sie)f(w)o(eiter:)827 1736 y Fe(A)864
1743 y Fb(v)q(;f)926 1736 y Fd(\016)c Fe(A)999 1743 y
Fb(w)q(;g)1068 1736 y Ff(=)j Fe(A)1157 1743 y Fb(v)q
Fc(+)p Fb(f)t Fc(\()p Fb(w)q Fc(\))p Fb(;f)t Fa(\016)p
Fb(g)302 1850 y Ff(\(b\))24 b(Zeigen)16 b(Sie,)f(da\031)i(durc)o(h)f
(die)f(V)l(erkn)1073 1852 y(\177)1072 1850 y(upfung)605
1954 y(\()p Fe(V)23 b Fd(\002)10 b Fe(GL)p Ff(\()p Fe(V)i
Ff(\)\))f Fd(\002)g Ff(\()p Fe(V)22 b Fd(\002)11 b Fe(GL)p
Ff(\()p Fe(V)h Ff(\)\))i Fd(!)g Fe(V)22 b Fd(\002)11
b Fe(GL)p Ff(\()p Fe(V)h Ff(\))696 2027 y(\(\()p Fe(v)r(;)c(f)d
Ff(\))p Fe(;)j Ff(\()p Fe(w)q(;)g(g)r Ff(\))o(\))14 b
Fd(7!)g Ff(\()p Fe(v)e Ff(+)f Fe(f)5 b Ff(\()p Fe(w)q
Ff(\))p Fe(;)j(f)17 b Fd(\016)11 b Fe(g)r Ff(\)\))391
2132 y(auf)16 b Fe(V)k Fd(\002)9 b Fe(GL)p Ff(\()p Fe(V)j
Ff(\))j(eine)f(Grupp)q(enstruktur)i(eingef)1336 2134
y(\177)1335 2132 y(uhrt)e(wird.)h(Zeigen)f(Sie)391 2192
y(w)o(eiter,)h(da\031)i(die)e(Abbildung)703 2296 y Fe(V)23
b Fd(\002)10 b Fe(GL)p Ff(\()p Fe(V)i Ff(\))i Fd(!)g
Ff(Bij)o(\()p Fe(A)p Ff(\))p Fe(;)8 b Ff(\()p Fe(v)r(;)g(f)d
Ff(\))13 b Fd(7!)g Fe(A)1418 2303 y Fb(v)q(;f)391 2401
y Ff(ein)j(injektiv)o(er)d(Grupp)q(enhomomorphism)o(us)i(ist.)1718
2501 y(\(6\))p eop
%%Page: 2 2
2 1 bop 222 274 a Ff(3.)24 b(Sei)f(\()p Fe(V)s(;)8 b(A)p
Ff(\))23 b(ein)f(a\016ner)i(Raum)e(mit)g Fe(n)p Ff(-dimensionalem)e(T)l
(ranslationsraum.)284 334 y(Zeigen)j(Sie,)f(da\031)j(f)661
336 y(\177)660 334 y(ur)f(Punkte)f Fe(p)931 341 y Fc(0)951
334 y Fe(;)8 b(:)g(:)g(:)16 b(;)8 b(p)1093 341 y Fb(n)1141
334 y Ff(folgende)23 b(Aussagen)1563 336 y(\177)1563
334 y(aquiv)m(alen)o(t)284 394 y(sind:)304 496 y(\(a\))i
Fe(p)415 503 y Fc(0)435 496 y Fe(;)8 b(:)g(:)g(:)16 b(;)8
b(p)577 503 y Fb(n)617 496 y Ff(bilden)15 b(ein)h(a\016nes)g
(Bezugssystem)f(v)o(on)h Fe(A)p Ff(.)302 577 y(\(b\))24
b(F)424 579 y(\177)423 577 y(ur)12 b(jeden)g(a\016nen)h(Raum)e(\()p
Fe(W)o(;)d(B)s Ff(\))k(und)h(je)f Fe(n)s Ff(+)s(1)h(Punkte)f
Fe(q)1498 584 y Fc(0)1518 577 y Fe(;)c(:)g(:)g(:)15 b(;)8
b(q)1657 584 y Fb(n)1694 577 y Fd(2)14 b Fe(B)391 637
y Ff(existiert)j(genau)i(eine)f(a\016ne)h(Abbildung)f
Fe(g)i Ff(:)d Fe(A)h Fd(!)f Fe(B)s Ff(,)h(so)i(da\031)f(f)1644
639 y(\177)1643 637 y(ur)g(alle)391 697 y Fe(i)14 b Ff(=)f(0)p
Fe(;)8 b(:)g(:)g(:)16 b(;)8 b(n)17 b Ff(die)e(Gleic)o(h)o(ung)g
Fe(g)r Ff(\()p Fe(p)1035 704 y Fb(i)1050 697 y Ff(\))f(=)f
Fe(q)1156 704 y Fb(i)1186 697 y Ff(gilt.)1718 799 y(\(4\))222
901 y(4.)44 b(\(a\))25 b(Sind)17 b Fe(U)536 908 y Fc(1)573
901 y Ff(und)g Fe(U)704 908 y Fc(2)741 901 y Ff(Un)o(terr)883
903 y(\177)883 901 y(aume)e(des)i(V)l(ektorraumes)e Fe(V)c
Ff(,)17 b(so)h(hei\031t)e Fe(V)28 b Ff(die)391 961 y(innere)14
b(direkte)f(Summe)f(v)o(on)j Fe(U)994 968 y Fc(1)1028
961 y Ff(und)g Fe(U)1157 968 y Fc(2)1177 961 y Ff(,)g(falls)f
(einerseits)f Fe(V)25 b Ff(=)14 b Fe(U)1654 968 y Fc(1)1682
961 y Ff(+)8 b Fe(U)1761 968 y Fc(2)391 1021 y Ff(ist)18
b(und)g(andererseits)f Fe(U)863 1028 y Fc(1)895 1021
y Fd(\\)12 b Fe(U)973 1028 y Fc(2)1009 1021 y Ff(=)17
b Fd(f)p Ff(0)p Fd(g)g Ff(gilt.)g(Zeigen)g(Sie,)g(da\031)h(in)g(dieser)
391 1081 y(Situation)e(die)g(Abbildung)853 1191 y Fe(f)j
Ff(:)13 b Fe(U)956 1198 y Fc(1)990 1191 y Fd(!)g Fe(V)5
b(=U)1143 1198 y Fc(2)1165 1191 y Fe(;)j(u)13 b Fd(7!)k
Ff(\026)-27 b Fe(u)391 1301 y Ff(ein)16 b(Isomorphism)o(us)e(v)o(on)i
(V)l(ektorr)1058 1303 y(\177)1058 1301 y(aumen)e(ist.)302
1382 y(\(b\))24 b(Sei)15 b Fe(U)22 b Ff(ein)15 b(Un)o(terraum)f(des)i
(V)l(ektorraumes)e Fe(V)e Ff(.)j(Zeigen)g(Sie,)g(da\031)i
Fe(V)27 b Ff(iso-)391 1443 y(morph)16 b(zur)630 1445
y(\177)630 1443 y(au\031eren)g(direkten)f(Summe)f Fe(U)i
Fd(\010)11 b Fe(V)5 b(=U)23 b Ff(ist.)264 b(\(4\))162
1557 y(Abgab)q(e:)20 b(Mitt)o(w)o(o)q(c)o(h,)e(9.5.2001,)i(9.00)g(Uhr,)
1027 1546 y(\177)1021 1557 y(Ubungsk)1208 1559 y(\177)1208
1557 y(asten)g(im)e(1.)h(Sto)q(c)o(k)h(des)f(Ma-)162
1617 y(thematisc)o(hen)14 b(Instituts.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF