?1?2(A?b)??4?2206??14?111r2?42?2?????3?206?r4?r121?r2?r1?????r?3r3得
所以
(2)
解②?①×2得 ③?① 得 得同解方程组
??322?31??322?31?32?123?38????123?38????14?236??0?32?1?5?(?1)?r2?r3??0?12?9?2????????0?25?62????14?236?4?23?01?2??16?921?292?r4?r3??0?32?1?5??????r3?3r20r4?2r2???00?4261???????0?25?62????001126????14?236???236??01?292??1401?292???001126?????r4?4r3???001126??,?00?4261????0007425????x1?4x2?2x3?3x4?6?? x2?2x3?9x4?2 x12x ?3?4?6?? 74x4?25??x1871??,?74?x?211?2,?74 ??x3?144?74,??x254?74.?① ?x1?2x2?2x3?2?2x1?5x2?2x3?4 ?② ?x1?2x2?4x3?6③ x2?2x3=0
2x3=4 ?x1?④ ?2x2?2x3?2? x2?2x3?0?? 2x3?4
⑤ ⑥ 由⑥得 x3=2, 由⑤得 x2=2x3=4,
由④得 x1=2?2x3 ?2x2 = ?10, 得 (x1,x2,x3)T=(?10,4,2)T. 2. 求下列齐次线性方程组的基础解系.
? x1?3x2?2x3?0,?(1) ? x1?5x2? x3?0, (2)
?3x?5x?8x?0;23?1? x1? x2?5x3? x4?0,? x? x?2x?3x?0,?1234 ??3x1? x2?8x3? x4?0,?? x1?3x2?9x3?7x4?0;? x1?2x2?2x3?2x4? x5?0,?? x1?2x2? x3?3x4?2x5?0, ?2x?4x?7x? x? x?0.2345?1? x1? x2?2x3?2x4?7x5?0,?(3) ?2x1?3x2?4x3?5x4 ?0, (4)
?3x?5x?6x?8x ?0;234?1【解】(1)
?x1?3x2?2x3?0,??x1?5x2?x3?0, ?3x?5x?8x?0.23?1?132??132??132?r3?2r2r2?r1?02?1?????02?1?
A??151??????r?3r??31????????358???0?42???000??得同解方程组
7?x??2x?3x??x3,32?12?x1?3x2?2x3?0???1 ?x?x,23?2x2?x3?0?2?x3?x3,?得基础解系为
T?7???2(2) 系数矩阵为
1?1?. 2?
?1?1A???3??1?1?0??0??0?11?13?12005?28?95?700?1??1?15?1??02?74?3?r3?r2r?r21???????????r4?2r23?3r1??1?r02?74r4?r1???7??04?148?
?1?4??r(A)?2.0??0?∴ 其基础解系含有4?R(A)?2个解向量.
3?x1???x3?x??2?2?x1?x2?5x3?x4?0???7?????x3?2???2x2?7x3?4x4?0?x3??x3???x?4???基础解系为
??3??x4???2???1??????2?7????2x4?x3?x4?? ??2??0???1???????1??x4??0????3???2????7?,?2??1?????0??
(3)
??1???2???. ?0????1??1122A??2345???3568?112r3?2r2?????010???000得同解方程组
7??11227?r2?2r1?0101?14?????0??r3?3r1???0???0202?21??
27?1?14??07???x1?x2?2x3?2x4?7x5?0,?x2?x4?14x5?0, ??7x5?0?x5?0.?取?
?x3??1??0????0?,?1?得基础解系为 x?4?????
(?2，0，1，0，0)T,(?1，?1，0，1，0).
(4) 方程的系数矩阵为
?12?2A??12?1???24?7?12r3?3r2?????00???002?1??12?22?1?r2?r1?0011?1?????3?2??r3?2r1???11???00?3?33??
?22?1?R(A)?2,11?1??000???x2??0??1??0??x???0?,?0?,?1?, ?4?????????1????0????0???x5???∴ 基础解系所含解向量为n?R(A)=5?2=3个
?x2???取x4为自由未知量 ????x5???3???2???4??0??1??0???????得基础解系 ?1?,?0?,??1?.
???????0??0??1???1????0????0??3. 解下列非齐次线性方程组.
?x1?x2?2x3?1,?2x1?x2?x3?x4?1,?2x?x?2x?4,??123(1) ? (2) ?4x1?2x2?2x3?x4?2,
?2x?x?x?x?1;?x1?2x2?3,1234???4x1?x2?4x3?2;?x1?x2?x3?x4?x5?7,x?2x?x?x?1,?1234?3x?2x?x?x?3x??2,??12345(3) ?x1?2x2?x3?x4??1, (4) ?
?x?2x?x?x?5;?x2?2x3?2x4?6x5?23,234?1??5x1?4x2?3x3?3x4?x5?12.【解】
（1） 方程组的增广矩阵为
?11?2?1(A?b)???1?2??41?11?0?3??00??00
21??11?0?324?r?2r21??????r3?r103?r4?4r1?0?3??42??0?321??1?01?r4?r3?22?2????????000????2?4??02?2?2?41?3001?2?r3?r2?????r4?r22???2?
21??22??12??00?
得同解方程组
x3?2,??x1?x2?2x3?1?2?2x3???3x?2x?2?x???2, ??223?3??x?23???x1?1?x2?2x3??1.(2) 方程组的增广矩阵为
?21?111??21?111?r3?r1?000?10?
(A?b)??42?212???????r2?2r1?????21?1?11???000?20??得同解方程组
?2x1?x2?x3?x4?1,??x4?0 ?x4?0,???2x4?0,?即
?2x1?x2?x3?1, ?x?0.?4令x1?x3?0得非齐次线性方程组的特解
xT=(0，1，0，0)T.
又分别取
?x2??1??0??x???0?,?1? ?3?????得其导出组的基础解系为
TT1??1???,1,0,0?;??2?∴ 方程组的解为
1??2???,0,1,0?,
?2??1??1???0??2??2??1?????x????k1?1??k2?0?.?0??0??1????????0?0??????0??k1,k2?R
1??1?2111??1?211??r2?r1??000?2?2?
(3) 1?21?1?1?????r3?r1????4??1?2115???0000?R(A)?R(A)∴ 方程组无解.