線性代數(shù)第二版答案.doc

線性代數(shù)第二版答案 篇一：工程數(shù)學(xué)線性代數(shù)課后答案同濟(jì)第五版 1 2 3 4 5 篇二：線性代數(shù)第二章答案 第二章矩陣及其運算 1?已知線性變換? ?x1?2y1?2y2?y3?x2?3y1?y2?5y3?x3?3y1?2y2?3y3 求從變量x1?x2?x3到變量y1?y2?y3的線性變換? 解由已知? ?x1?221?y1? ?x2?315?y2?x?323?y?2?3? ?y1?221?x1?7?49?y1?故y2?315?x2?63?7?y2?y?323?x?32?4?3?y3?2? ?y1?7x1?4x2?9x3 ?y2?6x1?3x2?7x3?y3?3x1?2x2?4x3 2?已知兩個線性變換?1 ?y1?3z1?z2?x1?2y1?y3 ?x2?2y1?3y2?2y3?y2?2z1?z3?y3?z2?3z3?x3?4y1?y2?5y3 求從z1?z2?z3到x1?x2?x3的線性變換? 解由已知 ?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?0?z1?1?z2?z?3?3? ?x1?6z1?z2?3z3 所以有?x2?12z1?4z2?9z3?x3?10z1?z2?16z3 ?111?123? 3?設(shè)A?11?1?B?1?24?求3AB?2A及ATB?1?11?051? ?111?123?111? 解3AB?2A?3?11?1?1?24?2?11?1?1?11?051?1?11? ?058?111?21322? ?3?0?56?2?11?1?2?1720?290?1?11?429?2? ?111?123?058? ATB?11?1?1?24?0?56?1?11?051?290? 4?計算下列乘積? ?431?7? (1)?1?23?2?570?1? ?431?7?4?7?3?2?1?1?35?解?1?23?2?1?7?(?2)?2?3?1?6?570?1?5?7?7?2?0?1?49? ?3? (2)(123)?2?1? ?3? 解(123)?2?(1?3?2?2?3?1)?(10)?1? ?2? (3)?1?(?12)?3? ?2?(?1)2?2?24?2? 解?1?(?12)?1?(?1)1?2?12?3?3?(?1)3?2?36? ?131?0?12?2140?(4)?1?31?1?134?40?2? ?131?0?12?6?78?2140?解?1?31?20?5?6?1?134?40?2? ?a11a12a13?x1?(5)(x1x2x3)a12a22a23?x2?aaa?132333?x3? 解 ?a11a12a13?x1?(x1x2x3)a12a22a23?x2?aaa?132333?x3? ?x1? ?(a11x1?a12x2?a13x3a12x1?a22x2?a23x3a13x1?a23x2?a33x3)?x2?x?3? 5?設(shè)A?22?a11x12?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3?1 ?12?B?1?13?0?問?2? (1)AB?BA嗎? 解AB?BA? 因為AB?3 ?44?BA?1?36?2?所以AB?BA?8? (2)(A?B)2?A2?2AB?B2嗎? 解(A?B)2?A2?2AB?B2? 因為A?B? ?2?22?5?2?2?25?2(A?B)2?2? 但2?814?1429?5?38?68?1A2?2AB?B2?411?812?3?0?1016?1527?4?所以(A?B)2?A2?2AB?B2? (3)(A?B)(A?B)?A2?B2嗎? 解(A?B)(A?B)?A2?B2? 因為A?B? ?2?22?A?B?0?05?2?1?22?02?06?(A?B)(A?B)?25?01?09? 10?28?38?而A2?B2?411?34?17? 故(A?B)(A?B)?A2?B2? 6?舉反列說明下列命題是錯誤的?（也可參考書上的答案） (1)若A2?0?則A?0? 解取A?0 ?0 ?1 ?01?則A2?0?但A?0?0?1?則A2?A?但A?0且A?E?0?(2)若A2?A?則A?0或A?E?解取A? (3)若AX?AY?且A?0?則X?Y? 解取 1A?0?0?X?11?Y?1?11?00?1?1? 則AX?AY?且A?0?但X?Y? 7?設(shè)A? 解 ?10?求A2?A3?Ak?1?10?10?10?A2?1?1?2?1?10?10?10?A3?A2A?2?1?1?3?1? 0?1?1Ak?k? ?10? 8?設(shè)A?0?1?求Ak?00? 解首先觀察 ?10?1A2?0?1?0?00?00?33?2A3?A2?A?0?3?00?44?3A4?A3?A?0?4?00?55?4A5?A4?A?0?5 ?00?