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1、1?O5|?5m5C?g.Jordan IO. I K JJIIJI11 ?84? w 4Draft? 2011c O?5“Linear Algebra(ES) O? or 1?O5|?5m5C?g.Jordan IO. I K JJIIJI12 ?84? w 4Draft1111?OOOXO1?K8yI(?:.?SN1. 1? ?a11a12 a1n a21a22 a2n. an1an2 ann?=Xj1,j2,jn(1)(j1,j2,jn)a1j1a2j2anjn.2. 1?5(J, ?)3. U1?m, .dn(“f).4. ?1? ?111 a1a2an. an11an12 an1n?
2、=Y16i ), ?, 8B, |?1?, E|, XA, |?A?.1?O5|?5m5C?g.Jordan IO. I K JJIIJI14 ?84? w 4Draft?;.K1?x 6= 0, A,B n ?, Oen?1?A =1 + x111 11 + x11 111 + x 1. 111 1 + x,B =x x.x x,C =?0A B10? .)|B| = xn. eO|A|.|A| =?1 + x111 11 + x11 111 + x 1. 111 1 + x?=?n + x n + x n + x n + x 11 + x11 111 + x 1. 111 1 + x?=(
3、n + x)?1111 1 1 + x11 111 + x 1. 111 1 + x?= (n + x)?1 1 1 1 0 x 0 0 0 0 x 0. 0 0 0 x?1?O5|?5m5C?g.Jordan IO. I K JJIIJI15 ?84? w 4Draft|A| = xn1(n + x).|?|A| =?11111 0 1 + x111 011 + x11 0111 + x 1. 0111 1 + x?=?11 1 1 1 1 x 0 0 0 1 0 x 0 0 1 0 0 x 0. 1 0 0 0 x?=?1 +n x1 1 1 1 0x 0 0 0 00 x 0 0 00
4、0 x 0. 00 0 0 x?= xn(1 +n x) = xn1(x + n).?|C| = |A|(1)(1+2+n)+(n+1+n+2+2n)|B1| = (1)nn + x x.1?O5|?5m5C?g.Jordan IO. I K JJIIJI16 ?84? w 4Draft21?Dn=? + 100 + 1.0 0 + .0. 000 + ?)U11mDn= ( + )Dn1 ?1100 0 + 1.0 0 + .0. 000 + ?= ( + )Dn1 Dn2.1?O5|?5m5C?g.Jordan IO. I K JJIIJI17 ?84? w 4Draft=Dn Dn1=
5、(Dn1 Dn2). Dn Dn1= n2(D2 D1) = n2(2+ 2+ 2 ) = n.n Dn Dn1= n.? ( )Dn= n+1 n+1.l? 6= , Dn=n+1n+1 .? = , ?Dn Dn1= n. ?Dn= n+ Dn1= n+ (n1+ Dn2) = 2n+ 2Dn2= = (n 1)n+ n1D1= (n + 1)n.1?O5|?5m5C?g.Jordan IO. I K JJIIJI18 ?84? w 4Draft3?A =2 1 0 1 2 0 0 0 1, ?B vABA= 2BA+ E, O|B|.)XJ)?B = (A 2E)1(A)1, 2, ?.
6、 ?(A 2E)BA= E.u, |A 2E|B|A| = 1.qJ?|A| = 3, dAA= |A|E = 3E, ?|A| = 9. d ?|A 2E| = 1 ,|B| =1 9.1?O5|?5m5C?g.Jordan IO. I K JJIIJI19 ?84? w 4Draft4?4 ?A = (,), B = (,). e|A| = 1, |B| = 2, O 1?|A + B|.)|A + B| =| + , + , + , + | = | + ,2( + + ), + , + | =2| + , + + , + , + | =2| + , + , + | = 2| + , +
7、,| =2| + ,| = 2(|,| + |,|) = 2(|A| + |B|) = 6.5e4 ?A ?A?O1,1,2,3, O1?|A1+ 2A|)dK?A?n). A?5-(?)? k=kA?u?,(=?),?kA? ?u?1?. d|A| = 6. ?A1+ 2A= A1(E + 2AA) = A1(E + 2|A|E) = 11A1.?|A1+ 2A| = | 11A1| = (11)4|A1| = 1146.1?O5|?5m5C?g.Jordan IO. I K JJIIJI110 ?84? w 4Draft6O1? ?a1b1+ xa1b2a1b3a1bn a2b1a2b2+
8、 xa2b3a2bn a3b1a3b2a3b3+ x a3bn. anb1anb2anb3 anbn+ x?,a1b1+ a2b2+ + anbn6= 0.)1?A?A = xE + B, B = (aibj) ?1 , ?1 (-?a.)0, ?dK , B 6= 0, u, B kn 1 A?0, ,kA?uTrB = a1b1+a2b2+ + anbn6= 0. l?, A kn 1 A?x, ,kA? x + TrB. l?|A| = xn1(x + a1b1+a2b2+ + anbn).)2dK?O.1?O5|?5m5C?g.Jordan IO. I K JJIIJI111 ?84?
9、 w 4Draft)3A =?xEnn+a1 a2. an(b1b2 bn)?|Em AmnBnm| = mn|En BnmAmn|?A = xn1(x + (b1b2 bn)a1 a2. an) = xn1(x + a1b1+a2b2+ + anbn).1?O5|?5m5C?g.Jordan IO. I K JJIIJI112 ?84? w 4Draft7?A 3 ?, X n, X,AX,A2X 5, A3X = 4AX 3A2X. O1?|2A2+ 3E|.)1dA3X = 4AX 3A2X ,A(A2+ 3A 4E)X = 0.dd, |A| = 0, K, XJA _, KA2X +
10、 3AX 4X = 0, X,AX,A2X 5g. l?0 A ?A?. qdA3X = 4AX 3A2X ? (A E)(A2X + 4X) = 0.n?|AE| = 0, =1 A ?A?. ?, 4 A ?A? . u2A2+ 3E ?nA?3,5,35. ,|2A2+ 3E| = 3 5 35 = 525.)2d?A(X,AX,A2X) = (AX,A2X,4AX3A2X) = (X,AX,A2X)0 00 1 04 0 1 3.5X,AX,A2X 5, ?A 0 00 1 04 0 1 3, l?(.1?O5|?5m5C?g.Jordan IO. I K JJIIJI113 ?84? w 4Draft8?n ?A vAA= I, |A| | = |A|2,?|A| = 1. u|A + E| = |A + AA| = |A|E + A| = |A + E|,l?|A + E| = 0.9?A m n ?, B n m ?, KA?m n , 7k|AB| 6= 0.B?m n , 7k|AB| = 0.C?m m ?. ye K?d:(1) Ax = k).(2) gAx = 0 ?)(x1,x2, ,xm)vx1b1+ x2b2+ + xmbm= 0
