計(jì)算機(jī)數(shù)學(xué)(A)第二次輔導(dǎo)課

1、1計(jì)算機(jī)數(shù)學(xué)（A）第二次輔導(dǎo)課安徽廣播電視大學(xué)教學(xué)處 吳和生2本章要點(diǎn) 線性方程組的有關(guān)概念， 增廣矩陣及增廣矩陣的階梯化， 線性方程組解的判定， 線性方程組的解法3一、 n元線性方程組的有關(guān)概念 1. n元線性方程組 由n個(gè)未知量，m道所有的未知量都是一次的方程組成的方程組稱為n元線性方程組。 習(xí)慣上我們用xj表示這些未知量，用aij表示它們的系數(shù)，用bj表示方程等號(hào)右邊的常數(shù)。4n元線性方程組的一般形式為mnmnmmnnnnbxaxaxabxaxaxabxaxaxa2211222221211121211111111221.nmmnnmxbaaxbaaxb系數(shù)矩陣未知變量52.齊次線性方程組

2、 如果常數(shù)項(xiàng)b1,b2,bm不全為0，則稱n元線性方程組為非齊次線性方程。 如果常數(shù)項(xiàng)b1,b2,bm全為0，則稱此線性方程組為齊次線性方程組。63. 消元法 1.用增廣矩陣A表示線性方程組AXB 2.把增廣矩陣中的系數(shù)矩陣部分用初等行變換化為階梯形矩陣 3.求出最后一個(gè)未知量的解，然后逐個(gè)方程地回代，求出其余的未知量 4.寫出線性方程組的解744213743242332352. 14321432143214321xxxxxxxxxxxxxxxx線性方程組解例253233121423471312414A解8(1,4)253233121423471312414124143121423471325

3、323A 解9124143121423471325323124140514480745501545 32-210(2,4)124140514480745501545124140154507455051448 1112414015450745505144812414015450039233000392433 121241401545003923300039243312414015450039233000013 13124140154500392330000131234234344:2445453923303xxxxxxxxxx 回代得144321:3,1,2,1:(1,2,1,3)xxxx 解得

4、或表示為12342343444343324454539233033392330396930,=-1xxxxxxxxxxxxxxx 代入1512341234123442.23932884xxxxxxxxxxxx 例 求解 300001111041111855501111041111488239113241111A解方程無解161234123412342342323883.3295234xxxxxxxxxxxxxxx例 解線性方程組171112321388:3219501234A解11123031420523401234 18111230123400121224005510 11123012340

5、5234031421911123012340011200551011123012340011200000 2011123012340011200000123423434232342xxxxxxxxx 211424342102xxxxxx1424341-22-xxxxxx224.行簡(jiǎn)化階梯矩陣 定義:滿足下列條件的階梯形矩陣稱為行簡(jiǎn)化階梯形矩陣 各非0行的第一個(gè)非0元素(首非0元)為1， 各首非0元所在列的其余元素均為0。2300000100001210022071例如171200012000001000001702000120000010000017020001200000100000XXX2

6、4四.行簡(jiǎn)化階梯矩陣 任意階梯形矩陣都可用初等行變換化為行簡(jiǎn)化階梯形矩陣。 因此，我們可以把增廣矩陣經(jīng)過初等行變換化為行簡(jiǎn)化階梯形矩陣。進(jìn)而求出線性方程組的一般解。2513423322438354. 44321432143214321xxxxxxxxxxxxxxxx解線性方程組例:145383114221113132413A解26145383114221113132413145380676900601807375A 27145380676900601807375145380676900103014114 28145380676900103014114145380606120010301012

7、429145380606120010301012145380000000103010123014538010120010300000 145380000000103010123114538010120010300000100110101200103000003210011010120010300000142434123xxxxxx 解為為自由變量331234123412341234352023505.74304157100 xxxxxxxxxxxxxxxx例 解線性方程組:3512023510174304157100A解3435122351A=1743415710174323513512415

8、710351743235135124157101743017137016117043232236174301713701611704323221743012001611704323223717430120016117043232217430120002170063223817430120002170063221743012000217000139174301200021700011740012000210000140174001200021000011740012000100001411740012000100001170001000010000142170001000010000110000

9、10000100001該齊次線性方程組只有零解43 .,0,:個(gè)解向量且每個(gè)基礎(chǔ)解系中含的基礎(chǔ)解系存在則定理rnAxnrArAnm例6:求一個(gè)齊次線性方程組的解0340222022432143214321xxxxxxxxxxxx44 000034210352010000342101221463046301221341122121221:2122313122312rrrrrrrrr系數(shù)矩陣解03420352432431xxxxxx45432431342352xxxxxx2413,cxcx2413212211342352cxcxccxccx1034350122214321ccxxxx46二.線性方

10、程組解的情況判定 線性方程組AXB有解的充分必要條件是它的系數(shù)矩陣的秩和增廣矩陣的秩相等，( )( ),( )( ),AAAXBr Ar Arrnrn即秩秩若線性方程組滿足則當(dāng)時(shí)線性方程組有唯一解當(dāng)時(shí)線性方程組有無窮多解47123412341234123425323223232542325328xxxxxxxxxxxxxxxx例25323223232542325328A解4825323223232542325328A解2532303000001000000549253230300000100000055025323030000010000005AA秩為3秩為451253230300000100

11、00000A=方程有無數(shù)解的情況521231231231231231231313123123123131:(1)(2)2020234234432243223253242(3)2023443203324xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 例判定下列方程組解的情況53無解解4)(,3)(47000181004110012129100181004110012155602650411001215203223441320121)1 (:ArArA54有唯一解解nArArA3)()(0000181004110012118100181004110012142562650411

12、0012142203223441320121) 2( :55有無窮多解nArArA2)()(0000000041100121246602055041100121243032013441320121)3(5612341234123412342.?7420253215854342xxxxxxxxxxxxxxxx例當(dāng) 為何值時(shí)，線性方程組有解 有解時(shí)求出它的解。30000000001259002471615270361527012590024712114345851235202471:A解57174200952100000000031742009521000000000035817420095210

13、000000000147109995210199900000000005914710999521019990000000000134234714999152999xxxxxx60齊次線性方程組解的判斷 齊次線性方程組AX有非零解的充要條件是r(A)n12312312301.00 xxxxxxxxx例 討論 的情況求方程的解11A= 1111解111111611111112110110112110110111101100(2)(1)22(1)(1) 11 12 62(1).=1當(dāng)時(shí)方程有無數(shù)解123123123000 xxxxxxxxx1 1 11111 1 10001 1 1000123123

14、0 xxxxxx 63(1).=-2當(dāng)時(shí)方程有無數(shù)解123123123202020 xxxxxxxxx21121112112111200064(2).=-2當(dāng)時(shí)方程有無數(shù)解21103312112100000012110101101100000013xx23xx65123123212312.xxxxxxxxx例討論 的情況求方程的解2222211111111111111110 11101166222222110 1110 11110 110 111 6722222110 110 111110 1100(1)(2)(1)(1) 6822110 1100(1)(2)(1)(1) 1231231231

15、23123(1)=-2,(2)=1,:x +x +x =1x +x +x =1x +x +x =1x +x +x =1x =1-x -x當(dāng)時(shí) 方程無解當(dāng)時(shí) 方程無數(shù)解 此時(shí)方程為6922110 1100(1)(2)(1)(1) (3)-2,1,當(dāng)且時(shí) 方程唯一解7022110 1100(1)(2)(1)(1) 22110 1100(2) (1)7122110 1100(2) (1)21101100(2)(1)7221101100(2)(1)211011(1)001273211011(1)001221110102(1)00127421110102(1)001221100-210102(1)00127512312321231xxxxxxxxx221231100-210102(1)001211(1)x =-,x =,x =2227612312321231xxxxxxxxx2212322221(1)-+222- +1+(1)- +1+21=122xxx77例 設(shè)非齊次線性方程組Ax=b的增廣矩陣484355121022121111000111,bA試求Ax=b的一般解.78解0