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D E /Fi 30 128 df<1238127C12FEA3127C123807077C8610>46
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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(5)162 394 y(Prof.)g(Dr.)h(B.)f(P)n(areigis)293 644
y Fh(Lineare)23 b(Algebra)g(und)h(analytisc)n(he)f(Geometrie)g(I)r(I)
222 758 y Fg(1.)h(Seien)15 b Ff(i;)8 b(j)16 b Fe(\024)e
Ff(n)p Fg(.)i(Die)g(P)o(erm)o(utation)e Ff(\034)990 765
y Fd(ij)1021 758 y Fg(,)h(die)h(durc)o(h)781 946 y Ff(\034)802
953 y Fd(ij)833 946 y Fg(\()p Ff(k)r Fg(\))d(=)963 829
y Fc(8)963 874 y(>)963 889 y(<)963 979 y(>)963 994 y(:)1007
876 y Ff(j)k Fg(:)d Ff(k)h Fg(=)f Ff(i)1007 948 y(i)g
Fg(:)f Ff(k)j Fg(=)e Ff(j)1007 1021 y(k)i Fg(:)e Ff(k)i
Fe(6)p Fg(=)d Ff(i;)8 b(j)284 1131 y Fg(hei\031t)14 b(die)g(T)l(ransp)q
(osition)i(v)o(on)e Ff(i)h Fg(und)f Ff(j)s Fg(.)g(Im)f(F)l(alle)h
Ff(j)i Fg(=)e Ff(i)8 b Fg(+)g(1)14 b(spric)o(h)o(t)f(man)h(v)o(on)284
1191 y(einer)h(Nac)o(h)o(bartransp)q(osition.)304 1293
y(\(a\))25 b(Zeigen)16 b(Sie,)f(da\031)i(jede)e(T)l(ransp)q(osition)j
(die)e(P)o(arit)1325 1295 y(\177)1325 1293 y(at)g Fe(\000)p
Fg(1)g(b)q(esitzt.)302 1374 y(\(b\))24 b(Zeigen)18 b(Sie,)f(da\031)i
(sic)o(h)e(jede)h(P)o(erm)o(utation)e(als)j(Pro)q(dukt)g(v)o(on)f(Nac)o
(h)o(bar-)391 1434 y(transp)q(ositionen)f(sc)o(hreib)q(en)f(l)963
1436 y(\177)963 1434 y(a\031t.)674 b(\(6\))222 1536 y(2.)24
b(Betrac)o(h)o(ten)14 b(Sie)i(die)g(Matrix)820 1592 y
Fc(\022)856 1632 y Fg(cos\()p Ff(\013)p Fg(\))42 b Fe(\000)8
b Fg(sin\()p Ff(\013)p Fg(\))859 1692 y(sin\()p Ff(\013)p
Fg(\))65 b(cos\()p Ff(\013)p Fg(\))1208 1592 y Fc(\023)304
1801 y Fg(\(a\))25 b(Bestimm)o(en)17 b(Sie)i(diejenigen)f(reellen)g
(Zahlen)h Ff(\013)p Fg(,)h(f)1365 1803 y(\177)1364 1801
y(ur)g(die)f(diese)g(Matrix)391 1861 y(einen)c(reellen)g(Eigen)o(w)o
(ert)g(b)q(esitzt.)302 1942 y(\(b\))24 b(Bestimm)o(en)13
b(Sie)j(die)g(k)o(omplexen)d(Eigen)o(w)o(erte)i(dieser)h(Matrix.)123
b(\(6\))222 2044 y(3.)24 b(Sei)c Ff(K)27 b Fg(=)22 b
Fb(Z)531 2051 y Fa(5)569 2044 y Fg(der)f(K)696 2046 y(\177)696
2044 y(orp)q(er)g(mit)e(5)j(Elemen)o(ten.)c(Betrac)o(h)o(ten)h(Sie)i
(die)f(3)15 b Fe(\002)f Fg(3-)284 2104 y(Matrix)911 2111
y Fc(0)911 2201 y(@)954 2151 y Fg(2)42 b(0)g(2)954 2211
y(3)g(0)g(2)954 2271 y(1)g(1)g(3)1110 2111 y Fc(1)1110
2201 y(A)284 2355 y Fg(mit)9 b(Ein)o(tr)479 2357 y(\177)479
2355 y(agen)i(aus)g Ff(K)t Fg(.)g(Bestimm)o(en)d(Sie)i(die)g(Eigen)o(w)
o(erte)g(dieser)g(Matrix)g(so)o(wie)284 2415 y(jew)o(eils)21
b(die)h(algebraisc)o(he)h(und)g(die)f(geometrisc)o(he)e(Vielfac)o
(hheit.)g(\(Hin)o(w)o(eis:)284 2475 y(Ein)c(Eigen)o(w)o(ert)f(ist)h
(4.\))991 b(\(6\))p eop
%%Page: 2 2
2 1 bop 222 274 a Fg(4.)24 b(Sei)15 b Ff(V)27 b Fg(der)15
b(Un)o(terraum)f(des)i(V)l(ektorraumes)e(aller)h(Abbildungen)g(v)o(on)g
Fb(R)e Fg(nac)o(h)284 334 y Fb(R)p Fg(,)20 b(der)k(v)o(on)f(den)h(F)l
(unktionen)f(sin\()p Ff(x)p Fg(\),)g(cos\()p Ff(x)p Fg(\))g(und)h(tan)q
(\()p Ff(x)p Fg(\))f(erzeugt)g(wird.)284 394 y(En)o(tsc)o(heiden)15
b(Sie,)g(ob)i(die)e(Basis)744 504 y Fe(\000)8 b Fg(sin\()p
Ff(x)p Fg(\))p Ff(;)g Fe(\000)g Fg(cos)o(\()p Ff(x)p
Fg(\))p Ff(;)g Fe(\000)g Fg(tan\()p Ff(x)p Fg(\))284
614 y(dieselb)q(e)15 b(Orien)o(tierung)g(b)q(esitzt)h(wie)f(die)h
(Basis)814 724 y(sin\()p Ff(x)p Fg(\))p Ff(;)8 b Fg(cos\()p
Ff(x)p Fg(\))p Ff(;)g Fg(tan\()p Ff(x)p Fg(\))284 834
y(\(Hin)o(w)o(eis:)17 b(Sie)h(brauc)o(hen)h(nic)o(h)o(t)e(zu)i(zeigen,)
f(da\031)i(es)f(sic)o(h)f(hierb)q(ei)g(um)g(Basen)284
895 y(handelt.\))1241 b(\(2\))162 1009 y(Abgab)q(e:)20
b(Mitt)o(w)o(o)q(c)o(h,)e(6.6.2001,)i(9.00)g(Uhr,)1027
998 y(\177)1021 1009 y(Ubungsk)1208 1011 y(\177)1208
1009 y(asten)g(im)e(1.)h(Sto)q(c)o(k)h(des)f(Ma-)162
1069 y(thematisc)o(hen)14 b(Instituts.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF