习题3.1
1．用消元法解下列线性方程组
?2x1?x2?3x3?1?2x1?x2?3x3?1?4x?2x?5x?4?1?23（1）?2x1 ?2x3?6 (2)?
2x?x?4x??1?4x?2x?5x?723?123?1??6x1?3x2?5x3?11?x1?x2?x3?x4?x5?1?x1?3x2?x3?x4?6?3x?2x?x?x?3x?0??2345(3)?3x1?x2?5x3?3x4?6（4）?1
x2?2x3?2x4?6x5?3?2x?x?2x?2x?8?234?1??5x1?4x2?3x3?3x4?x5?22．设线性方程组
?x1?x2?tx3?4?2??x1?tx2?x3?t ?x?x?2x??43?12t为何值时方程组无解? t为何值时方程组有解？有解时,求其解. 3．设线性方程组
?x1?x2?2x3?3x4?1?x?3x?6x?x?3?1234 ?3x?x?ax?15x?334?12??x1?5x2?10x3?12x4?b（1） a , b为何值时方程组有唯一解？ （2） a,b为何值时方程组无解？
（3） a,b为何值时方程组有无穷多解？并求其一般解.
习题3.2
1．设?1??1,1,?1,?2?，?2???2,1,0,1?，?3???1,?2,0,2?,求 （1）?1??2??3（2）2?1?3?2?5?3
2. 设 n 维向量?1?(1,0,?,0) , ?2?(0,1,?,0) , ?? ,?n?(0,0,?,1), 求 a1?1?a2?2???an?n.3. 设???2,?2, 0,4,2?, ???2,1, 3, ?1,1?，求向量? ,使??2??4?.
4．设?1??2,0,1?,?2??3,1,?1?满足2??3?1?4???2,求??．
5．设3??4??（211,,,2）, 2??3??（?1,2,31,）,求?,?.
习题3.3
1． 判断向量?能否由向量?1,?2,?3,?4线性表示,若可以,求出表达式. (1) ???1,1,?1,?1? , ?1??1,1,1,1?，?2??1,?1,1,?1?，?3??1,?1,?1,1?，?4??1,1,3,?1?(2) ???1,2,1,1? ,?1??1,1,1,1?，?2??1,1,?1,?1?，
?3??1,?1,1,?1?，?4??1,?1,?1,1?(3) ????4,?3,1,3? ?1??2,1,3,7?，?2???1,0,1,0?，?3??4,1,1,7?，?4???3,?1,0,?3?2. 设?1??2??0??3?????????47110 ?1???,?2???,?3???,?????0??1???1??b?????????23a???????4?(1) a, b 取何值时, ? 不能由?1,?2,?3线性表示;(2) a, b 取何值时, ? 能由?1,?2,?3唯一线性表示,写出该表达式;(3) a, b 取何值时, ? 能由?1,?2,?3线性表示且表达式不唯一,写出全体表达式.
3．判断下列向量组的线性相关性.
?4??1??2??4?????????120???????2?(1) ?1???，?2???，?3???，?4???10520?????????0??1??1??7??????????2???1??0??1?????????102???????3?(2) ?1???，?2???，?3???，?4????311?1?????????3??2??0??2??????????1??1??1???????1?1??????2?(3) ?1???，?2???，?3???215???????3??1??6????????2??1??0??4???????????1??1??3??4?(4) ?1???1?，?2??0?，?3??1?，?4??0?????????012???????4??3??2???1??1??????????4??5??6??4??2???????????2?675?????????3?(5) ?1???，?2???，?3???，?4???, ?5????3723?9???????????6??9??3??4??7????????????1??3???2??3??????????2??6??0??2?(6) ?1???，?2???，?3???，?4???1320?????????1??3??1???7??????????1??0??0???????01?????0?(7) ?1??0?，?2??0?，?3??1?
???????3??8??3???5??3??4???????4. 设向量组?1?a??2??3??4?????????1?2?a?3?4? ???? ?1?,?2?,?3?,?4??1??2??3?a??4??????????1??2??3??4?a?(1) a 为何值时, ?1,?2,?3,?4线性相关;（2）a 为何值时, ?1,?2,?3,?4线性无关.5．讨论向量组
?1??0??1???????12?????a??1??2?,?2??1?,?3??4?
???????2??5??b??1???1???1???????的线性相关性.
6．已知向量组?1,?,?i,?,?n线性无关,证明?1,?,k?i,?,?n(k?0)线性无关. 7．已知向量组?1,?2,?,?n线性无关, ?1??1,?2??1??2,?,?n??1??2????n, 证明: ?1,?2,?,?n线性无关.