%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: SS01Ub10.dvi %%Pages: 3 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips SS01Ub10.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.07.25:0022 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N 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03FEEA00F811207F9F16>I<38FF01FEA2381F003EAF147E14FE380F81BE3907FF3FC0EA 01FC1A167E951F>I<39FFE01FE0A2391F800700000F1306EBC00E0007130C13E000035B A26C6C5AA26C6C5AA2EB7CC0A2137F6D5AA26DC7FCA2130EA21B167F951E>I127 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 162 274 a Fj(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)734 b(F)-5 b(erien)n(blatt)162 394 y(Prof.)18 b(Dr.)h(B.)f(P)n(areigis)293 644 y Fi(Lineare)23 b(Algebra)g(und)h(analytisc)n(he)f(Geometrie)g(I)r (I)222 758 y Fh(1.)h(Es)g(sei)f Fg(f)p Ff(x)491 765 y Fe(1)510 758 y Ff(;)8 b(x)560 765 y Fe(2)579 758 y Ff(;)g(:)g(:)g(:)16 b(;)8 b(x)725 765 y Fd(n)748 758 y Fg(g)24 b Fh(eine)e(Menge)h(v)o(on)h (V)l(ektoren)e(eines)h(V)l(ektorraums)285 820 y(\177)284 818 y(ub)q(er)f(einem)d(K)586 820 y(\177)586 818 y(orp)q(er,)j(der)g (nic)o(h)o(t)e(die)h(Charakteristik)h(2)g(hat.)g(Ein)f(w)o(eiterer)284 878 y(V)l(ektor)15 b(sei)h Ff(y)r Fh(.)f(Zeigen)h(Sie:)f(Sind)h(die)g (Mengen)g(v)o(on)g(V)l(ektoren)371 988 y Fg(f)p Ff(x)424 995 y Fe(1)454 988 y Fh(+)11 b Ff(y)r(;)d(x)579 995 y Fe(2)609 988 y Fh(+)j Ff(y)r(;)d(:)g(:)g(:)15 b(;)8 b(x)829 995 y Fd(n)863 988 y Fh(+)j Ff(y)r Fg(g)27 b Fh(und)h Fg(f)p Ff(x)1152 995 y Fe(1)1183 988 y Fg(\000)11 b Ff(y)r(;)d(x)1309 995 y Fe(2)1339 988 y Fg(\000)i Ff(y)r(;)e(:)g(:)g(:)16 b(;)8 b(x)1560 995 y Fd(n)1594 988 y Fg(\000)j Ff(y)r Fg(g)284 1098 y Fh(linear)k(abh)497 1100 y(\177)497 1098 y(angig,)j(so)e(ist)g(auc)o(h)h Fg(f)p Ff(x)938 1105 y Fe(1)957 1098 y Ff(;)8 b(x)1007 1105 y Fe(2)1026 1098 y Ff(;)g(:)g(:)g(:)16 b(;)8 b(x)1172 1105 y Fd(n)1195 1098 y Fg(g)16 b Fh(linear)g(abh)1450 1100 y(\177)1450 1098 y(angig.)222 1200 y(2.)24 b(Sei)18 b Ff(V)31 b Fh(ein)19 b(endlic)o(hdim)o(ensionaler)d(V)l(ektorraum)1242 1202 y(\177)1241 1200 y(ub)q(er)k(dem)d(K)1503 1202 y(\177)1503 1200 y(orp)q(er)j Ff(K)k Fh(und)284 1260 y Ff(U)33 b Fg(\032)27 b Ff(V)36 b Fh(ein)24 b(ec)o(h)o(ter)f(Un)o(terv)o (ektorraum.)e(Bestimm)o(en)g(Sie)j(alle)g(Un)o(terv)o(ek-)284 1320 y(torr)365 1322 y(\177)365 1320 y(aume)15 b Ff(W)21 b Fg(\032)13 b Ff(V)f Fh(,)k(w)o(elc)o(he)e Ff(V)22 b Fg(n)11 b Ff(U)22 b Fh(en)o(thalten.)222 1422 y(3.)i(Im)14 b(a\016nen)i(Raum)g Ff(V)25 b Fh(=)14 b Fc(R)804 1404 y Fe(4)837 1422 y Fh(seien)h(folgende)h(vier)f(Punkte)h(gegeb)q(en:)423 1618 y Ff(v)447 1625 y Fe(1)481 1618 y Fh(=)532 1488 y Fb(0)532 1576 y(B)532 1606 y(B)532 1638 y(@)595 1528 y Fh(3)576 1588 y Fg(\000)p Fh(4)595 1648 y(1)595 1708 y(6)639 1488 y Fb(1)639 1576 y(C)639 1606 y(C)639 1638 y(A)691 1618 y Ff(;)35 b(v)764 1625 y Fe(2)798 1618 y Fh(=)849 1488 y Fb(0)849 1576 y(B)849 1606 y(B)849 1638 y(@)925 1528 y Fh(3)905 1588 y Fg(\000)p Fh(2)893 1648 y Fg(\000)p Fh(10)925 1708 y(0)980 1488 y Fb(1)980 1576 y(C)980 1606 y(C)980 1638 y(A)1032 1618 y Ff(;)h(v)1106 1625 y Fe(3)1139 1618 y Fh(=)1191 1488 y Fb(0)1191 1576 y(B)1191 1606 y(B)1191 1638 y(@)1254 1528 y Fh(2)1254 1588 y(0)1234 1648 y Fg(\000)p Fh(3)1254 1708 y(2)1298 1488 y Fb(1)1298 1576 y(C)1298 1606 y(C)1298 1638 y(A)1349 1618 y Ff(;)g(v)1423 1625 y Fe(4)1456 1618 y Fh(=)1508 1488 y Fb(0)1508 1576 y(B)1508 1606 y(B)1508 1638 y(@)1551 1528 y Fh(1)1551 1588 y(2)1551 1648 y(4)1551 1708 y(4)1576 1488 y Fb(1)1576 1576 y(C)1576 1606 y(C)1576 1638 y(A)1628 1618 y Ff(:)304 1822 y Fh(\(a\))25 b(Bestimm)o(en)9 b(Sie)i(die)g (Dimension)f(der)h(a\016nen)h(H)1295 1824 y(\177)1294 1822 y(ulle)e Ff(L)i Fh(v)o(on)f Fg(f)p Ff(v)1562 1829 y Fe(1)1582 1822 y Ff(;)d(:)g(:)g(:)15 b(;)8 b(v)1723 1829 y Fe(4)1742 1822 y Fg(g)p Fh(.)391 1882 y(\(Die)19 b(a\016ne)g(H)673 1884 y(\177)672 1882 y(ulle)f Ff(L)i Fh(ist)f(der)h(kleinste)e(a\016ne)h(Un)o(terraum)e(v)o(on)j Ff(V)11 b Fh(,)19 b(der)391 1942 y Fg(f)p Ff(v)440 1949 y Fe(1)459 1942 y Ff(;)8 b(:)g(:)g(:)16 b(;)8 b(v)601 1949 y Fe(4)620 1942 y Fg(g)16 b Fh(en)o(th)755 1944 y(\177)755 1942 y(alt.\))302 2023 y(\(b\))24 b(Bestimm)o(en)17 b(Sie)j(den)g(Durc)o(hsc)o(hnitt)f(v)o(on)h Ff(L)g Fh(mit)e(der)i(durc) o(h)g(die)g(Glei-)391 2083 y(c)o(h)o(ung)c(4)p Ff(x)584 2090 y Fe(1)615 2083 y Fh(+)11 b Ff(x)692 2090 y Fe(2)722 2083 y Fh(+)g Ff(x)799 2090 y Fe(3)830 2083 y Fg(\000)g Fh(2)p Ff(x)932 2090 y Fe(4)965 2083 y Fh(=)j Fg(\000)p Fh(6)j(gegeb)q(enen)f(Hyp)q(ereb)q(ene)f Ff(H)t Fh(.)222 2197 y(4.)24 b(Gegeb)q(en)16 b(sind)h(die)e(V)l(ektorr)827 2199 y(\177)827 2197 y(aume)f(und)j(linearen)e(Abbildungen)859 2319 y Ff(V)941 2291 y Fd(\013)912 2319 y Fg(\000)-9 b(!)14 b Ff(W)1102 2289 y Fd(\014)1073 2319 y Fg(\000)-9 b(!)14 b Ff(V)284 2429 y Fh(mit)g Ff(\014)g Fg(\016)d Ff(\013)j Fh(=)g(id)589 2436 y Fd(V)619 2429 y Fh(.)i(Bestimme)o(n)e (Sie)h(alle)h(Eigen)o(w)o(erte,)f(die)g(f)1430 2431 y(\177)1429 2429 y(ur)i Ff(\013)11 b Fg(\016)g Ff(\014)19 b Fh(m)1659 2431 y(\177)1659 2429 y(oglic)o(h)284 2490 y(sind.)p eop %%Page: 2 2 2 1 bop 222 274 a Fh(5.)24 b(Sei)13 b Ff(V)26 b Fh(ein)14 b(endlic)o(hdim)o(ensionaler)d(und)k Ff(W)21 b Fh(ein)14 b(v)o(on)g(0)g(v)o(ersc)o(hiedener)e(endlic)o(h-)284 334 y(dimensionaler)17 b(V)l(ektorraum)870 336 y(\177)869 334 y(ub)q(er)i(dem)f(K)1131 336 y(\177)1131 334 y(orp)q(er)i Ff(K)j Fh(und)d(sei)e(Hom)1603 341 y Fd(K)1637 334 y Fh(\()p Ff(V)s(;)8 b(W)f Fh(\))284 394 y(der)16 b(V)l(ektorraum)e(der)i (linearen)f(Abbildungen)g Ff(V)25 b Fg(!)14 b Ff(W)7 b Fh(.)15 b(Ein)h(Endomorphis-)284 455 y(m)o(us)f Ff(\013)f Fh(:)f Ff(V)25 b Fg(!)14 b Ff(V)28 b Fh(induziert)15 b(den)h(Endomorphism)o(us)542 546 y Ff(\013)573 525 y Fa(\003)606 546 y Fh(:)e(Hom)735 553 y Fd(K)770 546 y Fh(\()p Ff(V)s(;)8 b(W)f Fh(\))13 b Fg(!)h Fh(Hom)1092 553 y Fd(K)1127 546 y Fh(\()p Ff(V)s(;)8 b(W)f Fh(\))p Ff(;)23 b(f)d Fg(7!)13 b Ff(f)k Fg(\016)11 b Ff(\013)284 637 y Fh(Zeigen)k(Sie:)g Ff(\013)561 619 y Fa(\003)598 637 y Fh(hat)h(die)g(gleic)o(hen)f(Eigen)o(w)o(erte)g(wie)g Ff(\013)p Fh(.)284 716 y(Hin)o(w)o(eis:)f(Wie)h(steh)o(t)h(es)g(f)773 718 y(\177)772 716 y(ur)h Ff(W)901 702 y Fg(\030)902 718 y Fh(=)954 716 y Ff(K)t Fh(?)222 812 y(6.)24 b(In)16 b Fc(R)384 794 y Fe(4)417 812 y Fh(seien)f(die)h(V)l(ektoren)422 904 y Ff(a)448 911 y Fe(1)481 904 y Fh(=)e(\(1)p Ff(;)8 b Fh(1)p Ff(;)g Fh(1)p Ff(;)g Fh(1\))p Ff(;)36 b(a)809 911 y Fe(2)842 904 y Fh(=)14 b(\(1)p Ff(;)8 b Fh(2)p Ff(;)g Fh(2)p Ff(;)g Fg(\000)p Fh(1\))p Ff(;)36 b(a)1209 911 y Fe(3)1243 904 y Fh(=)13 b(\(0)p Ff(;)8 b Fh(1)p Ff(;)g Fh(1)p Ff(;)g Fg(\000)p Fh(2\))29 b(und)801 976 y Ff(x)13 b Fh(=)h(\(4)p Ff(;)8 b Fg(\000)p Fh(1)p Ff(;)g Fg(\000)p Fh(3)p Ff(;)g Fh(4\))284 1068 y(gegeb)q(en.)20 b(Sei)g Ff(P)28 b Fh(die)20 b(Orthogonalpro)s(jektion)h(\(bzgl.)f(des)h (Standard-Sk)m(alar-)284 1128 y(pro)q(dukts)c(in)f Fc(R)586 1110 y Fe(4)603 1128 y Fh(\))g(auf)h(den)f(v)o(on)g Ff(a)931 1135 y Fe(1)950 1128 y Ff(;)8 b(a)998 1135 y Fe(2)1017 1128 y Ff(;)g(a)1065 1135 y Fe(3)1101 1128 y Fh(erzeugten)15 b(Un)o(terraum)f Ff(U)5 b Fh(.)304 1224 y(\(a\))25 b(Bestimm)o(en)13 b(Sie)j Ff(y)f Fh(:=)f Ff(P)7 b(x)p Fh(.)302 1300 y(\(b\))24 b(Geb)q(en)17 b(Sie)e(eine)g(Orthonormalbasis)i(aus)g(Eigen)o(v)o (ektoren)d(v)o(on)i Ff(P)24 b Fh(an.)222 1396 y(7.)44 b(\(a\))25 b(Zeigen)16 b(Sie,)f(da\031)i(durc)o(h)f(die)f(linearen)h (Gleic)o(h)o(ungssysteme)878 1488 y Ff(X)918 1495 y Fe(1)949 1488 y Fg(\000)11 b Fh(2)p Ff(X)1063 1495 y Fe(2)1095 1488 y Fh(+)g Ff(X)1184 1495 y Fe(3)1218 1488 y Fh(=)j(0)902 1560 y Ff(X)942 1567 y Fe(1)974 1560 y Fh(+)d Ff(X)1063 1567 y Fe(2)1094 1560 y Fg(\000)g Ff(X)1184 1567 y Fe(3)1218 1560 y Fh(=)j(1)391 1652 y(und)891 1743 y Ff(X)931 1750 y Fe(1)962 1743 y Fh(+)d(2)p Ff(X)1075 1750 y Fe(2)1107 1743 y Fh(+)g Ff(X)1196 1750 y Fe(3)1230 1743 y Fh(=)i(0)866 1816 y(2)p Ff(X)930 1823 y Fe(1)962 1816 y Fh(+)e(3)p Ff(X)1075 1823 y Fe(2)1107 1816 y Fh(+)g Ff(X)1196 1823 y Fe(3)1230 1816 y Fh(=)i(0)391 1907 y(zw)o(ei)i(windsc)o(hiefe)g (Geraden)h Ff(g)974 1914 y Fe(1)1011 1907 y Fh(bzw.)f Ff(g)1147 1914 y Fe(2)1184 1907 y Fh(in)h Fc(R)1280 1889 y Fe(3)1313 1907 y Fh(gegeb)q(en)g(w)o(erden.)302 1983 y(\(b\))24 b(Bestimm)o(en)13 b(Sie)j(den)g(Abstand)h(v)o(on)f Ff(g)1128 1990 y Fe(1)1164 1983 y Fh(und)h Ff(g)1285 1990 y Fe(2)1305 1983 y Fh(.)307 2059 y(\(c\))24 b(Bestimm)o(en)19 b(Sie)j(den)f(Cosin)o(us)i(des)f(Wink)o(els)f(zwisc)o(hen)f Ff(g)1532 2066 y Fe(2)1575 2059 y Fh(und)i(einer)391 2119 y(zu)16 b Ff(g)479 2126 y Fe(1)515 2119 y Fh(parallelen)g(Gerade)g (durc)o(h)g(den)g(Ursprung.)222 2215 y(8.)24 b(Es)17 b(sei)e(die)h(reelle)e(Matrix)785 2353 y Ff(A)p Fh(\()p Ff(a)p Fh(\))e(=)950 2253 y Fb(0)950 2343 y(@)994 2293 y Fh(1)42 b(1)g(3)12 b(+)f Ff(a)994 2353 y Fh(0)42 b(3)g Ff(a)10 b Fg(\000)h Fh(1)994 2413 y(0)42 b(0)g(2)12 b(+)f Ff(a)1237 2253 y Fb(1)1237 2343 y(A)284 2501 y Fh(v)o(orgegeb)q(en.)p eop %%Page: 3 3 3 2 bop 304 274 a Fh(\(a\))25 b(Geb)q(en)17 b(Sie)f(die)h(Jordansc)o (he)g(Normalform)d(v)o(on)j Ff(A)p Fh(\()p Ff(a)p Fh(\))f(in)h(Abh)1589 276 y(\177)1589 274 y(angigk)o(eit)391 334 y(v)o(on)f Ff(a)g Fh(an.)302 415 y(\(b\))24 b(Geb)q(en)17 b(Sie)e(die)h Ff(A)p Fh(\(3\)-in)o(v)m(arian)o(ten)f(eindimensionalen)f(T)l(eilr)1550 417 y(\177)1550 415 y(aume)g(an.)222 517 y(9.)24 b(Bestimm)o(en)9 b(Sie)i(die)h(Jordansc)o(he)g(Normalform)e(einer)h(reellen)f(\()p Ff(n)r Fg(\002)r Ff(n)p Fh(\)-Matrix,)284 577 y(deren)16 b(s)436 579 y(\177)436 577 y(am)o(tlic)o(he)d(Zeilen)i(gleic)o(h)f(\(1) p Ff(;)8 b Fh(2)p Ff(;)g(:)g(:)g(:)17 b(;)8 b(n)p Fh(\))16 b(sind.)197 679 y(10.)25 b(Es)19 b(sei)g(f)445 681 y(\177)444 679 y(ur)f(einen)g(Endomorphism)o(us)g Ff(A)g Fh(:)g Fc(R)1152 661 y Fe(7)1187 679 y Fg(!)g Fc(R)1294 661 y Fe(7)1329 679 y Fh(das)i(Minimalp)q(olynom)284 739 y Ff(\026)p Fh(\()p Ff(\025)p Fh(\))14 b(=)g Ff(\025)473 721 y Fe(3)493 739 y Fh(.)304 841 y(\(a\))25 b(W)l(elc)o(he)15 b(Dimension)g(k)m(ann)h(der)g(Kern)g(v)o(on)g Ff(A)g Fh(hab)q(en?)302 922 y(\(b\))24 b(Man)19 b(geb)q(e)g(f)637 924 y(\177)636 922 y(ur)g(jede)f(m)847 924 y(\177)847 922 y(oglic)o(he)e(Dimension)h(v)o(on)i(Kern)e Ff(A)i Fh(alle)f(Jordan-)391 982 y(sc)o(hen)e(Normalformen)d(v)o(on)j Ff(A)g Fh(an.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF