线性代数-北京邮电大学出版社-戴赋祥

?2?(?9,?8,?13)用这个基线性表示.

【解】设

A?(?1,?2,?3),B?(?1,?2),

又设

?1?x11?1?x21?2?x31?3,?2?x12?1?x22?2?x32?3,

即

?x11(?1,?2)?(?1,?2,?3)??x21??x31记作 B=AX.

则

x12?x22??, x32???1235(A?B)???1110???0327?1235?0327???002?2?9??12r2?r1?????03?8?????13???03?9??1作初等行变换???????0?13?????4???0354527001001?9?r2?r3?????17??r2?r3?13??

23?3?3???1?2??因有A?E,故?1,?2,?3为R3的一个基，且

?23?(?1,?2)?(?1,?2,?3)?3?3?,

?????1?2??即

?1?2?1?3?2??3,?2?3?1?3?2?2?3.

（B类）

1.A

2.B 3.C 4.D

5.a=2，b=4 6.abc≠0

7.设向量组α1,α2,α3线性相关,向量组α2,α3,α4线性无关,问: (1) α1能否由α2,α3线性表示?证明你的结论. (2) α4能否由α1,α2,α3线性表示?证明你的结论.

解：(1)由向量组α1，α2，α3线性相关，知向量组α1, α2, α3的秩小于等于2,而α2, α3, α4线性无关，所以α2, α3线性无关，故α2, α3是α1, α2, α3的极大线性无关组，所以α1能由α2, α3线性表示.

（2）不能.若α4可由α1，α2，α3线性表示,而α2，α3是α1，α2，α3的极大线性无关组,所以α4可由α2，α3线性表示.与α2，α3，α4线性无关矛盾.

8.若α1,α2,…,αn,αn+1线性相关,但其中任意n个向量都线性无关,证明:必存在n+1个全不为零的数k1,k2,…,kn,kn+1,使

k1α1+k2α2+…+kn+1αn+1=0.

证明:因为α1，α2，…，αn，αn+1线性相关，所以存在不全为零的k1，k2，…，kn，kn+1使k1α1+k2α2+…+kn+1αn+1=0

若k1=0，则k2α2+…+kn+1αn+1=0，由任意n个向量都性线无关，则k2=…=kn+1=0,矛盾.从k1≠0,同理可知ki≠0,i=2, …,n+1,所以存在n+1个全不为零的数k1,k2,…,kn,kn+1,使k1a1+k2a2+…+kn+1an+1=0.

9. 设A是n×m矩阵,B是m×n矩阵,其中n＜m,E为n阶单位矩阵.若AB=E,证明:B的列向量组线性无关. 证明：由第2章知识知，秩A≤n,秩B≤n，可由第2章小结所给矩阵秩的性质，n=秩E≤min{秩A，秩B}≤n，所以秩B=n，所以B的列向量的秩为n，即线性无关.

习题四

（A类）

1. 用消元法解下列方程组.

?x1?4x2?2x3?3x4?6,?x1?2x2?2x3?2,?2x?2x?4x?2,??124(1) ? (2) ?2x1?5x2?2x3?4,

?x?2x?4x?6;?3x1?2x2?2x3?3x4?1,23?1??x1?2x2?3x3?3x4?8;【解】(1)

?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?