(高等教育出版社)线性代数课后习题1-4作业答案

高等教育出版社线性代数课后习题1-4作业答案

?1?3 (3)?2?3??1?3?2?33534?4?4?2?235343?1?? 0??1??3?1?(下一步? r?3r? r?2r? r?3r? )

213141

0??1???1?3 解 ?2?3??1?3?2?3?4?4?2?2?1?0 ~?0?0??1?0 ~?0?0??1?0 ~?0?0??13?43?0?48?8?(下一步? r?(?4)? r?(?3) ? r?(?5)? )

234

0?36?6?0?510?10???1000?10003111?4?2?2?23?2?(下一步? r?3r? r?r? r?r? )

123242

2?2??02?3?1?22?? 000?000??3. 已知两个线性变换

??x1?2y1?y3 ?x2??2y1?3y2?2y3?

??x3?4y1?y2?5y3 解 由已知

??y1??3z1?z2?y2?2z1?z3? ??y3??z2?3z3求从z1? z2? z3到x1? x2? x3的线性变换?

?x1??201??y1??201???31 ?x2????232??y2????232??20?x??415??y??415??0?1??2?????3????613??z1? ??12?49??z2??

??10?116??z????3???x1??6z1?z2?3z3所以有?x2?12z1?4z2?9z3?

??x3??10z1?z2?16z3

0??z1?1??z2? ?z?3???3?高等教育出版社线性代数课后习题1-4作业答案

4. 试利用矩阵的初等变换? 求下列方阵的逆矩阵?

?321? (1)?315??

?323????321100??321100? 解 ?315010?~?0?14?110?

?323001??002?101??????3203/20?1/2??3007/22?9/2? ~?0?1011?2?~?0?1011?2?

?002?10??1????001?1/201/2??1007/62/3?3/2? ~?010?1?12?

?001?1/201/2????72?3??632?

故逆矩阵为??1?12??

?11??0?2?2???3?20?1??0221? (2)??

1?2?3?2??0121???

?3?20?11000??02210100? 解 ?

1?2?3?20010??01210001????1?2?3?20010??01210001? ~?

049510?30??02210100????1?2?3?20010??01210001? ~?

001110?3?4??00?2?1010?2???高等教育出版社线性代数课后习题1-4作业答案

?1?2?3?20010??0121000?1 ~? 001110?3?4??000121?6?10????1?200?0100 ~?0010?0001??1?0 ~?0?0?01000010?10?12?11`?11?2?2?0?1? 36??6?10???1?0故逆矩阵为??1?2?011?2?4?0010?1? 0?1?136?121?6?10??1?2?4?10?1?? ?136?1?6?10???021?123?? 求X使XA?B? 5. (2)设A??2?13?? B????2?31??33?4????? 解 考虑ATXT?BT? 因为

?02?312?r?1002?4? (AT, BT)??2?132?3?~ ?010?17??

?13?431??001?14??????2?4?所以 XT?(AT)?1BT???17??

??14???2?1?1? 从而 X?BA?1????474????9. 求作一个秩是4的方阵? 它的两个行向量是

(1? 0? 1? 0? 0)? (1? ?1? 0? 0? 0)?

解 用已知向量容易构成一个有4个非零行的5阶下三角矩阵?

高等教育出版社线性代数课后习题1-4作业答案

?1?1?1?0?0?0?100000100000100?0?0?? 0?0??此矩阵的秩为4? 其第2行和第3行是已知向量?

?1?23k?12. 设A???12k?3?? 问k为何值? 可使

?k?23??? (1)R(A)?1? (2)R(A)?2? (3)R(A)?3?

k?1?23k?r?1?1?????? 解 A??12k?3~ 0k?1k?1?k?23??00?(k?1)(k?2)????? (1)当k?1时? R(A)?1? (2)当k??2且k?1时? R(A)?2? (3)当k?1且k??2时? R(A)?3? P106/ 1.已知向量组

A? a1?(0? 1? 2? 3)T? a2?(3? 0? 1? 2)T? a3?(2? 3? 0? 1)T? B? b1?(2? 1? 1? 2)T? b2?(0? ?2? 1? 1)T? b3?(4? 4? 1? 3)T? 证明B组能由A组线性表示? 但A组不能由B组线性表示? ?0?1 证明 由 (A, B)??2?3??1r? ~ ?00?0?031?60200430122301204??1r?1?24?~0 111??0?213???0031?24?32204? 1?6?15?7?2?8?17?9??1?24??1r??15?7?~0 5?1525??0?1?35???0031?24?1?6?15?7? 041?35?00000??知R(A)?R(A? B)?3? 所以B组能由A组线性表示? 由