《線性代數(shù)與解析幾何》復(fù)習(xí)要點(diǎn)

1、 1221()12( 1)nnjjjjjnjNa aa2211()12( 1)nniiNiii nia aa 運(yùn)算前提條件定義性質(zhì)加法A + BA與B是同類型的對應(yīng)元素相加A + B = B + A; (A + B) + C = A + (B + C);A + O = A; A + (A) = O數(shù)乘kAk是一個(gè)數(shù)用k乘A的每一個(gè)元素k(lA) = (kl)A; (k + l)A = kA + lA;k(A + B) = kA + kB; (1)A = A乘法ABA的列數(shù) = B的行數(shù)(aij)ml(bij)ln = (cij)mn cij = (AB)C = A(BC); A(B+C) =A

2、B+AC;(A+B)C =AC+BC; (kA)B = k(AB)冪 AmA是方陣, m是正整數(shù)A1 = A, Ak+1 = AkAAkAl = Ak+l; (Ak)l = Akl轉(zhuǎn)置AT無(aij)ml T = (aji)lm(AT)T = A; (A + B)T = AT + BT;(kA)T = kAT; (AB)T = BTAT多項(xiàng)式f(A)A是一個(gè)方陣,f(x) = asxs + a1x + a0f(A) = asAs +a1A+a0IA = ( )f(A) = f( ) ,A = ( ), f(A) = O f( ) = 0行列式|A|A是一個(gè)方陣, |A1| = |A|1逆矩陣A

3、1A是一個(gè)方陣且|A|0若AB = BA = I則B = A1唯一性, (A1)1 = A, (A1)m = (Am)1,(AT)1 = (A1)T, (kA)1 = k1A1, (AB)1 = B1A1, 滿秩, 特征值01nikkjka b: : : : 3 1 01 3 00 0 4A,2/ 1 0 2/ 12/ 1 0 2/ 1 0 1 0 Q.4 0 00 4 00 0 2T1 AQQAQQ;1 1 232111311 1 1|,2222332 .6/ 1 3/ 1 2/ 16/ 1 3/ 1 2/ 1 6/2 3/ 1 0 ),(321 qqqQ msssmmaaaaaaaaaA2

4、12221212111, : mnmmnnccccccccc 212222111211 msmmssaaaaaaaaa 212222111211 snssnnbbbbbbbbb 212222111211 mnnnmmcccccccccC212221212111, : 212222111211snssnnbbbbbbbbb 212222111211mnmmnnccccccccc mnmmnnccccccccc 212222111211 msmmssaaaaaaaaa 212222111211 snssnnbbbbbbbbb 212222111211 mnmmnnccccccccc 2122221

5、11211 msmmssaaaaaaaaa 212222111211 snssnnbbbbbbbbb 212222111211 mnmmnnaaaaaaaaaA 212222111211 mnmmnnbbbbbbbbbB 212222111211 mnmmnnaaaaaaaaaA 212222111211 mnmmnnbbbbbbbbbB 2122221112110 10 1A0 00 1B0 01 1C0 00 1B 0 37702 3520 432143214321xxxxxxxxxxxx 1 3 7 7 2 3 5 21 1 1 1 0 0 0 07/4 7/5 1 07/3 7/2 0

6、 1,1 0 7/47/3 ,01 7/57/22 1 ).,( ,1 0 7/47/301 7/57/2112 14321Rccccxxxx ,11 ,1143 xx).,( ,1 1 7/ 1 7/ 11 1 7/97/511214321Rccccxxxx ,7/ 1 7/ 1 ,7/97/521 xx,1 1 7/ 1 7/ 1 ,1 1 7/97/521 1 3 7 7 2 3 5 21 1 1 1 0 0 0 0 0 1 3 4 1 0 2 5,23 1 0 ,5 40 1 21 ).,( ,23 1 0 5 40 1 21214321Rccccxxxx 21421325 34 xxxxxx 22222211100pprrnfyyyyyy 且規(guī)范形是唯一的且規(guī)范形是唯一