电大《高等代数专题研究》形成性考核册作业答案.doc

专业好文档高等代数专题研究作业参考答案高等代数专题研究作业1一、单项选择题：1-5：BCBDB二、填空题1、交换。2、不等价、等价。3、，且是A到B的双射。4、具有下面性质的自然数的任何集合M满足：如果，则。则M含有一切自然数，即。5、对于一个与自然数有关的命题T，若i：若n=1时命题T正确；ii：假设命题T对n<k正确，就能推出命题T对n=k正确。则命题T对一切自然数正确。三、计算题1、解：到的映射一共有个，它们是：，2、解：，3、解：1）在G中，并且，可表为两个不相交的轮换的乘积：。2），3）四、证明题1、证明：2、证明：则于是由a与b惟一确定的（即不会得出以上不同的结果），且为实数，所以“”是一个代数运算。，所以，即“”满足结合律。3、证明：当n=2时，因此命题对n=2正确。当n=4时，因此命题对n=4正确。同理可推出命题对，都正确（s为任意自然数），所以命题对无穷多个自然数成立。设命题对n=k正确，令，则，由归纳假设命题对n=k正确，所以，所发，即，命题对n=k-1也正确，由反归纳法原理知，命题对一切自然数成立。4、当n=2时，上述不等式成立，假设，则于是对一切的自然数n来说，。五、简述题1、答：，给予证明如下：任取，且，则是单射。任取，若为奇数，则有，使与之对应；若为偶数，则有，使与之对应，所以有是满射。所以是从Z到N的双射。2、答：空集合的幂集不是空集合。应为。高等代数专题研究作业2一、单项选择题：1-5：DACCB二、填空题：1、 2、 3、4、 5、三、计算题1、解：所以原不等式的解集为。2、解：，即。其中当且仅当，且成立，解得，所以当时，取极大值，。3、解：这是一个求具有约束条件的极值问题，由于它有三个变量，因而不能用消元法来解，但，只有当时等式成立。所以只有当时，取最小值。四、证明题1、证明：，因都是正数，上式变为，得证。2、证明：令，再令，得的一元二次方程：，由于，所以，所以，即。3、证明：因为是等差数列，则，则均值不等式，得，又：，所以，所以，故结论得证。五、简述题1、答：设函数在某区间上定义，对于区间上的任意两点，都有，其中，则称在该区间上是下凸函数。2、答：比较法、综合法、分析法、数学归纳法、反证法、换元法、放缩法。高等代数专题研究作业3一、单项选择题：1-5：BDDAC二、填空题1、1，3，5，7 2、如果d是a与b的公因式，且有，均有。 3、代数4、1 5、-4，2（重根）三、计算题1、证：1）若，则，且，故是有单位元素1的数环，因而是整环。2）为中全部可逆元素。为奇素）为中全部不可约元素。2、解：是的可逆元素。，是的可逆元素。因此，是的全部可逆元素。四、证明题1、证明：首先是整环，零理想是主理想，设是的任一非零理想，是中次数最低的多项式，则对任意有，使，其中或的次数<的次数，由知，若则的次数<的次数，这与是中次数最低的多项式矛盾，故必有，从而，这就证明了是由生成的主理想。2、证：若之中有零或单位，易见结论成立。不妨设都既非零也非单位，因为，所以有，将都分解为不可约元素的乘积，若非单位也将其分解：，则，由因式分解的惟一性，每个都与等式左边的一个因子相伴，因为，所以不与任何一个相伴，适当调整因子的次序，不妨设分别与相伴，于是可知。3、证：由可知，因是本原多项式，所以，由上第2题结论知：。4、证：设，若从代数观点出发，则它们相应系数有以下关系：，显然它们在任意点的函数值也相同，即从函数论观点出发。反之，若从函数论观点出发，则，这时域中所有元素都是的根。但是是一个次数不超过的多项式，在中至多有个根，而前述有无限多个根，这个矛盾证明必有，即从代数观点有。五、简述题1、答：定义：设是一个整环，如果中每一个不等于的非单位元素均可写成：，其中是不可约元素，并且如果还有，其中也是不可约元素，则必有，且适当调整的顺序后，有，则称是因式分解惟一环。2、答：定理：任何实系数次多项式至少有一个复数根。高等代数专题研究作业4一、单项选择题：1-5：BDCAA二、填空题1、 2、 3、4、， 5、三、计算题1、解：把辆小轿车视为一辆，与辆大卡车排队有种方法，而小轿车又有种停放方法，所以一共有种停放方法。2、解：用相同元素的重复排列公式：，不同的摆法有：种。3、解：展开合并同类项后共有：展开后每一项都是5次多项式，它的不同项实际上是从6个元素中取5个元素的方法数项，而的系数为：从3元素中取2个a，2个b，1个c，即为，所以的系数为30。4、解：设表示能被整除而不大于2000的自然数集合，这时，根据定理：5、解：用递推公式：，对楼梯作归纳：当，只有一个台阶，只有一种走法；当，可以一步一阶，也可一步两阶，；当，可一步一阶，也可一步两阶一阶或一阶两阶，；，。6、解：设，1）至少参加一项比赛的人数：2）只参加四百米比赛的人数：。四、证明题1、证明：因，所以，所以。令，上式化为，所以，当时，有。2、证明：因，两边对求导一次，得到，上式两边乘以，再对求导，得，令，整理得。五、简述题答：抽屉原理：如果把件东西装入个抽屉，则至少有1个抽屉里的东西不少于2件。抽屉原理：设都是正整数；和进有件东西放进个抽屉，则第1个抽屉至少有件东西，或第2个抽屉至少有件东西，或第个抽屉至少有件东西，其中至少有一条必成立。"If we don't do that it will go on and go on. We have to stop it; we need the courage to do it."His comments came hours after Fifa vice-president Jeffrey Webb - also in London for the FA's celebrations - said he wanted to meet Ivory Coast international Toure to discuss his complaint.CSKA general director Roman Babaev says the matter has been "exaggerated" by the Ivorian and the British media.Blatter, 77, said: "It has been decided by the Fifa congress that it is a nonsense for racism to be dealt with with fines. You can always find money from somebody to pay them."It is a nonsense to have matches played without spectators because it is against the spirit of football and against the visiting team. It is all nonsense."We can do something better to fight racism and discrimination."This is one of the villains we have today in our game. But it is only with harsh sanctions that racism and discrimination can be washed out of football."The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's Premier League match at Chelsea on Sunday."I am going to be at the match tomorrow and I have asked to meet Yaya Toure," he told BBC Sport."For me it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's complaint. In a statement the Russian side added: "We found no racist insults from fans of CSKA."Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet ( Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.Below, the Earth becomes the concrete bottom of a swimming pool that he wants to land on, but not too hard. Still, he'll be traveling fast, so despite the distance, it will not be like diving into the deep end of a pool. It will be like he is diving into the shallow end.Skydiver preps for the big jumpWhen he jumps, he is expected to reach the speed of sound - 690 mph (1,110 kph) - in less than 40 seconds. Like hitting the top of the water, he will begin to slow as he approaches the more dense air closer to Earth. But this will not be enough to stop him completely.If he goes too fast or spins out of control, he has a stabilization parachute that can be deployed to slow him down. His team hopes it's not needed. Instead, he plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).In order to deploy this chute successfully, he will have to slow to 172 mph (277 kph). He will have a reserve parachute that will open automatically if he loses consciousness at mach speeds.Even if everything goes as planned, it won't. Baumgartner still will free fall at a speed that would cause you and me to pass out, and no parachute is guaranteed to work higher than 25,000 feet (7,620 meters).cause there