高等代数葵花宝典v2.0.pdf-资料库

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8 „ 1 :n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . A, B nz . 1.1 “o( . . . 1.2 UkfmCX . . . 1.3 . 1.4 § AX = XB ? . . 1.5 r A L« A ı“. . 1.6 1.7 A«? . . . . . . . . . . . . . . . . A ı“ Jordan IO/ . . . . . . . . . 1.7.1 1 . . . 1.7.2 . . 1.7.3 " . . 1.7.4 Gram . . 1.7.5 . . 1.7.6 Hilbert . 1.7.7 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 7 . . 8 . 11 . 14 . 18 . 20 . 20 . 21 . 22 . 22 . 23 . 24 . 25 2 ﬂK8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.1 ﬂK . 2.2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 . 32 1

1 :n 1.1 “o( {“· r(AB) 6 min{r(A), r(B)}, r(A + B) 6 r(A) + r(B), øp 2y†. y†“{k—CA{, {3ıŒ ·B. A^{·e¡ø“: XJ ( ) M = A 0 C B øp A, B ·, @o r(M ) > r(A) + r(B). y†g··kr A, B z⁄IO/: P1AQ1 = Er, P2BQ2 = Es, @o ) ( )( )( ( ) 0 P1 0 P2 0 Er D Es ,–^ Er ZK D c r , ^ Es ZK D s 1, M C⁄øf 0 P2CQ1 Es 0 Q1 0 Q2 A 0 C B = ( = )  Er 0 0 0 0 ∗ 0 0 Es Er  M = ø(. 3y†k’“, ˜g··lOØu, z⁄n‰en '‹, 5A^¡ø(. Frobenius “y† A ·Œ F m × n , B · n × k , C · k × s , K y†: ( ) r(AB) + r(BC) 6 r(ABC) + r(B) ) ( ) 0 −ABC BC B AB 0 B BC −−−−−−−−−−−−−−−→ 1mƒ– ¡C \1 ( −−−−−−−−−−−−−−−→ 11ƒ– ¡A \11 0 −ABC B 0 1

CHAPTER1. :n 1.1. “o( ’(. 5¿øp¢·r¢g·“X”5. 3œAT·l ( ) ABC 0 B 0 u5C. /“c¡nU⁄øf. pnK ﬂﬂﬂKKK 1: (2005ﬁ˘) A ·Œ F n m5C, ƒy A3 = I ⇔ r(I − A) + r(I + A + A2) = n y†: 5¿3 u(x), v(x) ƒ u(x)(1 − x) + v(x)(1 + x2 + x) = 1, ˜ 2n ( I − A ( ) 0 I − A u(A)(I − A) I + A + A2 0 ) ( −−−−−−−−−−−−−−−→ 1ƒ– u(A) \1 ( I − A I + A + A2 ) ) 0 I ) I I + A + A2 ( 0 A3 − I 0 I A3 − I 0 0 I + A + A2 −−−−−−−−−−−−−−−→ 11ƒ– v(A) \1 −−−−−−−−−−−−−−−→ 1ƒ– A ¡ I \1 −−−−−−−−−−−−−→ me I + A + A2 ⁄–ø· n = A3 − I = 0, øy†. (2006ﬁ˘) A, B · n , y† ﬂﬂﬂKKK 2: r(A − ABA) = r(A) + r(I − BA) − n ( ) 0 A BA I − BA y†: ( ) ( 0 A 0 I − BA −−−−−−−−−→ 1\1 A A BA I −−−−−−−−−−−−−−−→ 11ƒ– ¡A \11 ) −−−−−−−−−−−−−−−→= 11ƒ– B \11 ) ) ( ( A − ABA 0 BA I A − ABA 0 I 0 −−−−−−−−−−−−−−−−→ 1mƒ– ¡BA \1 –C·—C, –C. l“⁄Æ. 2

CHAPTER1. :n 1.1. “o( ﬂﬂﬂKKK 3: (2007ﬁ˘) A, B · n v AB = BA. ƒy r(A) + r(B) > r(A + B) + r(AB) y†: X ·g5§| AX = 0 )m, Y ·g5§| BX = 0 )m, Z ·g5§| ABX = BAX = 0 )m, W ·g5 §| (A + B)X = 0 )m, @o•k X ⊂ Z, Y ⊂ Z. l X + Y ⊂ Z. X ∩ Y ⊂ W . dŒœ“, dimX + dimY = dimX ∩ Y + dim(X + Y ) 6 dimW + dimZ l n − r(A) + n − r(B) 6 n − r(A + B) + n − r(AB) = r(A) + r(B) > r(A + B) + r(AB). 5: øK8–^—C5, ·w,A{{’. ÆSK ﬂﬂﬂKKK 1: A, B ·Œ F n v AB = BA = 0, r(A) = r(A2), ƒy r(A + B) = r(A) + r(B) e¡'O^C{A{y†, §’3uØ r(A) = r(A2) ø^|^. Cy†: g·: y† r(A + B) > r(A) + r(B) =. ^P{, l ( A + B 0 0 0 ( ) ) B ∗ 0 A u, ˇL—Cz/X , (. 3Cck5¿:: (1)du A2 · A 5|, ⁄– r(A) = r(A2) ‘† A | A2 |·d, · A – A2 5L«5. αi · A 1 i , @o5§| A2X = αi k) Xi, - n × n P = (X1, X2,··· , Xn), @o A2P = A. (2)g( r(A) = r(A2) k r(A) = r(A2) = r(A3) = ··· y†: 3 Frobenius “ r(B) + r(ABC) > r(AB) + r(BC) 3

CHAPTER1. :n 1.1. “o( ¥- B = C = A J(. e¡?1C:( ) A + B 0 0 0 −−−−−−−−−−−−−−→ 1mƒ– A \1 −−−−−−−−−−−−−−−→ 1mƒ– ¡P \1 ) −−−−−−−−−−−−−−→= 11ƒ– A \11 ) A + B A2 A3 ( ( A2 B A2 0 A3 ( ) A + B 0 0 A2 l r(A + B) > r(B) + r(A3) = r(B) + r(A), y. A{: ø{g·5gØ^ r(A2) = r(A) ,«). ˜ky† e¡n: r(A2) = r(A) ⇒ V = KerA ⊕ ImA. y†: η1, η2, ··· , ηn¡r · KerA |˜, Aε1, Aε2, ··· , Aεr · ImA | ˜. •I‘† η1,··· , ηn¡r, Aε1,··· , Aεr ⁄ V |˜=. ø KerA + ImA ·, Œ· n, n y†. " c1,··· , cn ƒ n¡r∑ ciηi + ciAεj = 0 (∗) @o3>^ A ^: i=1 · A2X = 0 AX = 0 )m·, ⁄– A2( ciεj) = 0 A( ciεj) = 0 = £(∗)“ j=1 j=1 r∑ r∑ r∑ r∑ n¡r∑ j=1 j=1 ciAεj = 0 ciηi = 0 i=1 4

CHAPTER1. :n 1.1. “o( ⁄– ci ·0, ny. ⁄Ky†: Jy KerA ImA · B Cfm. B 3 ImA ·0. ⁄–3˜ η1, ··· , ηn¡r, Aε1, ··· , Aεr e A, B /X ( ) ( A = 0 0 0 A1 B = B1 0 0 0 y3 r(A + B) = r(A) + r(B) ·Øw,. 5: XJØ$E|vG{, øK8–w5. –b A · Jordan IO/, @o r(A) = r(A2) ‘† A 0AØA Jordan ‹· . l A /X ) ) A1 0 0 0  A1 (  = ) 0 ... 0 ( 0 B1 0 B2 øp A1 ·_. AB = BA = 0 ‘† B /X (ø^. ˇ A1 vkA0, ⁄– A 7,·OØ. [„1.4) øp A1B1 = B1A1 = 0. du A1 _, ⁄– B1 = 0. ø y†. ﬂﬂﬂKKK 2: A, B n , ƒy r(A + B) = r(A) + r(B) ¿^·3_ P , Q ƒ ( ( ) ) P AQ = P BQ = Er 0 0 0 0 0 0 Es ¥ r, s 'O· A, B , r + s 6 n. y†: y75. –k P , Q ƒ ( ) P AQ = Er 0 0 0 ⁄–b A ·IO/, e¡y†–^—CUC A /G, r B C⁄/G. ( ) B = B1 B2 B3 B4 5

) 1.1. “o( CHAPTER1. :n ( A + B = ( s = r(B) > r Er + B1 B2 B4 ) B3 > r(A + B) − r = s 1·ˇl A + B ¥c r ı~ r. ⁄– lkC n ⁄–k1C ( B2 B4 ) ) B2 B4 r ( = r(B) ( B = B1 B2 B3 B4 ( C−−−−→ ) 0 B2 0 B4 r ( 0 0 B3 B4 ) = r(B) ( B = B1 B2 B3 B4 1C−−−−→ ) ) ) B = B1 B2 B3 B4 −−−−−−→ 1, C ( ) ( ) → ( 0 0 B3 B4 0 0 0 B4 (  0 0 → )  0 0 0 0 0 0 Es 5¿øCUC A /G. du1C–p, ⁄– B –UYzIO/, UC A /G. B = B1 B2 B3 B4 ø⁄y†. 0 0 0 B4 6

1.2. UkfmCX CHAPTER1. :n 1.2 UkfmCX ˜ F kˆ¡ı(’X·Œ), @o F m V U§kı fmCX. ·3fm V1, V2,··· ,Vm ƒ V ⊆ V1 ∪ V2 ∪ ··· ∪ Vm. y†: y{: V ⊆ V1 ∪ V2 ∪ ··· ∪ Vm · V ⁄kfmCX¥(? m − 1 fmUCX V ). @oØ?1 6 i 6 m, Vi $ V1 ∪ V2 ∪ ··· ∪ Vi¡1 ∪ Vi+1 ∪ ··· ∪ Vm. ˜K m ·4gæ. lz Vi kgC“k”. V1 k α V2 k β, e¡y5| λα + β ¥kˆ¡ıÆu V1, V2,··· ,Vm ¥? . ø·ˇ λα + β oÆu V1, g i > 2 XJ λ1 λ2 ƒ λ1α + β, λ2α + βÆ3 Vi ¥, @o (λ1α + β) − (λ2α + β) = (λ1 − λ2)α Æ3 Vi ¥, α ‰´gæ! ⁄–z Vi ¥ık/X λα + β. ˜kˆ¡ı , l8{ λα + β | λ ∈ F }¥kˆ¡ı, ⁄–7kˆ¡ıU V1,··· ,Vm CX. V ·k/, k{’y†: ε1, ε2,··· ,εn · V |˜, αi = ε1 + iε2 + ··· + in¡1εn d1“£ˆ¡S α1, α2,··· , αn,··· ¥? n ⁄ V | ˜, kıfmU„¥kı, lkˆ¡ı αi UCX. e¡ø(4›. XJm V 5C A Aı“4ı“, @o3 v ∈ V ƒ v, Av, ··· , An¡1v · V |˜. y†: f (x) · A Aı“, m(x) · A 4ı“. (, V ¥? v 4z"ı“ mv(x) A 4ı“ m(x). · m(x) kkı ˇf, ⁄– v H V , mv(x) kkıU m1(x), m2(x),··· ,mk(x). ˜Vi = {v ∈ V m, ⁄–7k, Vi ƒ V = Vi, @ok m(x) = mi(x), = f (x) = mi(x). ·‘ 3 V ¥ v ƒ v 4z"ı“· A Aı“. @o n v, Av, ··· , An¡1v 7,·5ˆ’£o⁄, l§⁄ V |˜. 3'¥ø v ¡“”. ﬂﬂ mi(A)v = 0}, @o V Æu V1,··· ,Vk ¿, l V1,··· ,Vk U·f 7

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