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1、(1)?cosxsinx sinxcosx?n ;(2)?11 11?n ;(3)a1 a1 a1 a1 an.?: (1)?cosnxsinnx sinnxcosnx? ;(2) (2)n?cosn 4sinn 4sinn 4cosn 4?n ;(? (1) ?.)2. ?: ? n ? I.?: ? A = (aij) ? A ? Eij?, ? AEij= EijA. ? ? AEij? j ? A ? i ? 0 ?EijA ? i ? A ? j ? ? 0 ? A ? 0? aii= ajj? A ? ?. 3. ? A1B ? CA1, ?A =450 231 273, B =4
2、5010 2311 2793, C =450 231 279 237.?: A1B = 1 2424060175 0244892 10354598,CA1= 1 242400 0242 9621618 11219222.4. ? A,B Mn, ?: adj(AB) = adj(B)adj(A).?(1)? A,B ?(2)? x F ?, ? x ?, A xI ? B xI ?, ? ? x ?adj(A xI)(B xI) = adj(B xI)adj(A xI).? i ? j ? x ? ? ?x ? x = 0 ? adj(AB) = adj(B)adj(A).5. ?: ? A,
3、 ? r(AA) = r(AA) = r(A).?. ? ? yAA= 0 ?, ? (AA)= 0? (A)(A)= 0, ? A ? 0? A = 0. ? yAA= 0 ? yA = 0 ? ?.6. ?: ? n ? A, ? r(An) = r(An+1).4? r(Ai+1) r(Ai), ? A,A2, ,An+1? 2 ? As? At?. ? s 0, ac b2.?ab bc? . ? ?ab bc? a 0, ac b2.31. ? V = acost + bsint, ?a,b? ?. ? f,g V , ?(f,g) = f(0)g(0) + f(2)g(2).? (
4、f,g) ? V ?, ? h(t) = 3cos(t + 7) + 4sin(t + 9) ?.? (f,g) ? V ?.h(t) = 3cos(t+7)+4sin(t+9) ? 5.?acost + bsint ?a2+ b2.?32. ? Rx2?(f,g) =1?1f(x)g(x)dx.(1) ? 1,t,t2?; (2) ? f(x) = 1 x + x2? g(x) = 1 4x 5x2?.?1? G =202/3 02/30 2/302/5; (2) 0.33. ? ai,1 i n ?, xi,yi?, ?(n?i=1aixiyi)2 (n?i=1aix2i)(n?i=1ai
5、y2i).? x = (x1, ,xn)T,y = (y1, ,yn)T Rn, ?(x,y) =n?i=1aixiyi? (x,y) ? Rn?. ? ?.34. (1)? C ? R ? 2 ?. ? C ?, ? i ? 1 + i ? C ?, ? (2)? R3?, ? e1,e1+ e2,e1+ e2+ e3? ?. ? e2? e3? ?1?. ?2? e1,e1+e2,e1+e2+e3? (e1,e1) = 1,(e1,e2) = 1,(e1,e3) = 0,(e2,e2) = 2,(e2,e3) = 1,(e3,e3) = 2. ?(x1,x2,x3),(y1,y2,y3)
6、= x1y1+ 2x2y2+ 2x3y3 (x1y2+ x2y1+ x2y3+ x3y2).1035. ?34?.? ?.36. ?Cauchy-Schwarz?.?Cauchy-Schwarz? = 0 ? ?= 0. ? ? t ? ( t, t) 0, ?(,) 2(,t) + t2(,) 0.? t = (,)/(,) ?. ?(+)2= 2+2+2(,) 2+2+2| (?Cauchy-Schwarz?).37.? R4?, ? 1= (1,0,1,1)T, 2= (2,1,0,3)T? 3= (1,1,1,1)T?.? (?)?1= (1,0,1,1)T/3, 2= (7,3,1,
7、8)T/123,3= (3,54,23,20)T/3854.38. ? V ? ? ?d(,) = | |.? d(,) ? V ?, ?: (d1) ?: d(,) = d(,); (d2) ?: d(,) 0, ? = ; (d3) ?: d(,) + d(,) d(,).?.39. ?2? V ? 1,2, ?A =?54 45? .? V ? 1,2?.? 1,2? 1= 1+ 2,2= 1 2? 1= (1+ 2)/18,2= (1 2)/2 ?.? 1,2? ? 1= 1/5,2= (41 52)/45 ?.?40. ? n ? V ? 1,2, ,n, ? A. ? , V ?
8、? x ? y. (1)? (,) = xTA y. ?, ? V ?, (,) = xTAy. (2)? (1) ?.?1? B = (1,2, ,n), ? = Bx ? = By ? (,) ?.(2) ? 1,2, ,n? V ? G = (i,j). ? , ? ? u ? v. ? 1,2, ,n? 1,2, ,n? P. ? x = Pu,y = Pv. ? i= BPi,1 i n, ? Pi? P ? i ?. ?(i,j) = (BPi,BPj) = PTiAPj,1 i,j n.11? G = (i,j) = PTAP.? (,) = uTG v = uTPTAP v =
9、 (Pu)TAPv = xTA y. ? (1) ? ?.41. ? V = Mn(R) ? Mn(C). ? A = (aij),B = (bij) V . (1)? (A,B) = tr(AB) ? V ?; (2)? (1) ?, ? A ? ? (3)?: ? Eij,1 i,j n ? V ?; (4)? M2(R) ?.?1?.?2?(A,A) = tr(AA) =?n i,j=1|aij|2.(3) (Eij,Ekl) = tr(EijEkl) = tr(EijElk). ? EijElk= jlEik, ? tr(EijElk) = 1 i = k,j = l? tr(EijE
10、lk) = 0. ? Eij,1 i,j n ? V ?.?4?I/2,?01 10? /2,?01 10? /2,?10 1? /2. (?.)42. ? V = R2? (?). ? 1= (1,1)T, 2= (1,1)T? 1= (0,2)T,2= (6,12)T? V ?. ? j? k?(1,1) = 1,(1,2) = 15,(2,1) = 1,(2,2) = 3.(1) ?; (2) ? V ?.?1? (1,1) = 2,(1,2) = 1,(2,2) = 2.(1,1) = 2,(1,2) = 12,(2,2) = 135.(2) 1= (1+ 2)/6,2= (1 2)/2 ? V ?.?.?43. ? n ? V ? 1,2, ,n. ? C, ?(1,2, ,n) = (1,2, ,n)C? 1,2, ,n? V ?.?. ? A ? ? ATA ? Q ? QTATAQ = D ?. ? D1QTATAQD1= I, ? (AQD1)T(AQD1) = I. ? AQD1?, ? ? R. ? A = RDQT= (RDRT)RQT. ? D ? RDRT? R,Q ? ? RQT?.? 1,2, ,n? V ?1,2, ,n? P? ? (1,2, ,n) = (1,2, ,n)P. ? C ? Q ?12? P = CQ.
