??(a1? a2? ?????? an)T ?V1? ??(b1? b2? ?????? bn)T ?V1? 有 a1?a2? ???????an?1? b1?b2? ???????bn?1?
从而 (a1?b1)?(a2?b2)? ???????(an?bn) ?(a1?a2? ???????an)?(b1?b2? ???????bn)?2? 所以 ????(a1?b1? a2?b2? ?????? an?bn)T?V1?
37? 试证? 由a1?(0? 1? 1)T? a2?(1? 0? 1)T? a3?(1? 1? 0)T所生成的向量空间就是R3. 证明 设A?(a1? a2? a3)? 由
011|A|?101??2?0?
110知R(A)?3? 故a1? a2? a3线性无关? 所以a1? a2? a3是三维空间R3的一组基, 因此由a1? a2? a3所生成的向量空间就是R3.
38? 由a1?(1? 1? 0? 0)T? a2?(1? 0? 1? 1)T所生成的向量空间记作V1,由b1?(2? ?1? 3? 3)T? b2?(0? 1? ?1? ?1)T所生成的向量空间记作V2, 试证V1?V2.
证明 设A?(a1? a2)? B?(b1? b2)? 显然R(A)?R(B)?2? 又由 ?1?1 (A, B)??0?0?10112?1330??1r?1? ~0 ?1??0??1???01?1002?3000?1?? 0?0??知R(A? B)?2? 所以R(A)?R(B)?R(A? B)? 从而向量组a1? a2与向量组b1? b2等价? 因为向量组a1? a2与向量组b1? b2等价? 所以这两个向量组所生成的向量空间相同? 即V1?V2.
39? 验证a1?(1? ?1? 0)T? a2?(2? 1? 3)T? a3?(3? 1? 2)T为R3的一个基, 并把v1?(5? 0? 7)T? v2?(?9? ?8? ?13)T用这个基线性表示. 解 设A?(a1? a2? a3)? 由
123|(a1, a2, a3)|??111??6?0?
032知R(A)?3? 故a1? a2? a3线性无关? 所以a1? a2? a3为R3的一个基. 设x1a1?x2a2?x3a3?v1? 则
??x1?2x2?3x3?5??x1?x2?x3?0? ??3x2?2x3?7解之得x1?2? x2?3? x3??1? 故线性表示为v1?2a1?3a2?a3? 设x1a1?x2a2?x3a3?v2? 则
??x1?2x2?3x3??9??x1?x2?x3??8? ??3x2?2x3??13解之得x1?3? x2??3? x3??2? 故线性表示为v2?3a1?3a2?2a3?
40? 已知R3的两个基为
a1?(1? 1? 1)T? a2?(1? 0? ?1)T? a3?(1? 0? 1)T? b1?(1? 2? 1)T? b2?(2? 3? 4)T? b3?(3? 4? 3)T? 求由基a1? a2? a3到基b1? b2? b3的过渡矩阵P? 解 设e1? e2? e3是三维单位坐标向量组? 则
?111? (a1, a2, a3)?(e1, e2, e3)?100??
?1?11????1 (e1, e2, e3)?(a1, a2, a3)?1?1??1于是 (b1, b2, b3)?(e1, e2, e3)?2?1?10?11?0?? 1??23?34? 43???1?1?111??123? ?(a1, a2, a3)?100??234??
?1?11??143?????由基a1? a2? a3到基b1? b2? b3的过渡矩阵为
?111??123??234? P??100??234???0?10??
?1?11??143???10?1???????
?1