電大《高等代數(shù)專題研究》形成性考核冊作業(yè)答案

1、高等代數(shù)專題研究作業(yè)參考答案高等代數(shù)專題研究作業(yè)1一、單項(xiàng)選擇題：1-5：bcbdb二、填空題1、交換。2、不等價(jià)、等價(jià)。3、，且是a到b的雙射。4、具有下面性質(zhì)的自然數(shù)的任何集合m滿足：如果，則。則m含有一切自然數(shù)，即。5、對(duì)于一個(gè)與自然數(shù)有關(guān)的命題t，若i：若n=1時(shí)命題t正確；ii：假設(shè)命題t對(duì)nk正確，就能推出命題t對(duì)n=k正確。則命題t對(duì)一切自然數(shù)正確。三、計(jì)算題1、解：到的映射一共有個(gè)，它們是：，2、解：，3、解：1）在g中，并且，可表為兩個(gè)不相交的輪換的乘積：。2），3）四、證明題1、證明：2、證明：則于是由a與b惟一確定的（即不會(huì)得出以上不同的結(jié)果），且為實(shí)數(shù)，所以“”是一個(gè)代

2、數(shù)運(yùn)算。，所以，即“”滿足結(jié)合律。3、證明：當(dāng)n=2時(shí)，因此命題對(duì)n=2正確。當(dāng)n=4時(shí)，因此命題對(duì)n=4正確。同理可推出命題對(duì)，都正確（s為任意自然數(shù)），所以命題對(duì)無窮多個(gè)自然數(shù)成立。設(shè)命題對(duì)n=k正確，令，則，由歸納假設(shè)命題對(duì)n=k正確，所以，所發(fā)，即，命題對(duì)n=k-1也正確，由反歸納法原理知，命題對(duì)一切自然數(shù)成立。4、當(dāng)n=2時(shí)，上述不等式成立，假設(shè)，則于是對(duì)一切的自然數(shù)n來說，。五、簡述題1、答：，給予證明如下：任取，且，則是單射。任取，若為奇數(shù)，則有，使與之對(duì)應(yīng)；若為偶數(shù)，則有，使與之對(duì)應(yīng)，所以有是滿射。所以是從z到n的雙射。2、答：空集合的冪集不是空集合。應(yīng)為。高等代數(shù)專題研究作業(yè)

3、2一、單項(xiàng)選擇題：1-5：daccb二、填空題：1、 2、 3、4、 5、三、計(jì)算題1、解：所以原不等式的解集為。2、解：，即。其中當(dāng)且僅當(dāng)，且成立，解得，所以當(dāng)時(shí)，取極大值，。3、解：這是一個(gè)求具有約束條件的極值問題，由于它有三個(gè)變量，因而不能用消元法來解，但，只有當(dāng)時(shí)等式成立。所以只有當(dāng)時(shí)，取最小值。四、證明題1、證明：，因都是正數(shù)，上式變?yōu)椋米C。2、證明：令，再令，得的一元二次方程：，由于，所以，所以，即。3、證明：因?yàn)槭堑炔顢?shù)列，則，則均值不等式，得，又：，所以，所以，故結(jié)論得證。五、簡述題1、答：設(shè)函數(shù)在某區(qū)間上定義，對(duì)于區(qū)間上的任意兩點(diǎn)，都有，其中，則稱在該區(qū)間上是下凸函數(shù)。2、

4、答：比較法、綜合法、分析法、數(shù)學(xué)歸納法、反證法、換元法、放縮法。高等代數(shù)專題研究作業(yè)3一、單項(xiàng)選擇題：1-5：bddac二、填空題1、1，3，5，7 2、如果d是a與b的公因式，且有，均有。 3、代數(shù)4、1 5、-4，2（重根）三、計(jì)算題1、證：1）若，則，且，故是有單位元素1的數(shù)環(huán)，因而是整環(huán)。2）為中全部可逆元素。為奇素）為中全部不可約元素。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、證明題1、證明：首先是整環(huán)，零理想是主理想，設(shè)是的任一非零理想，是中次數(shù)最低的多項(xiàng)式，則對(duì)任意有，使，其中或的次數(shù)的次數(shù)，由知，若則的次數(shù)的次數(shù)，這與是中次數(shù)最低的多項(xiàng)式矛盾，故必有，從

5、而，這就證明了是由生成的主理想。2、證：若之中有零或單位，易見結(jié)論成立。不妨設(shè)都既非零也非單位，因?yàn)?，所以有，將都分解為不可約元素的乘積，若非單位也將其分解：，則，由因式分解的惟一性，每個(gè)都與等式左邊的一個(gè)因子相伴，因?yàn)?，所以不與任何一個(gè)相伴，適當(dāng)調(diào)整因子的次序，不妨設(shè)分別與相伴，于是可知。3、證：由可知，因是本原多項(xiàng)式，所以，由上第2題結(jié)論知：。4、證：設(shè)，若從代數(shù)觀點(diǎn)出發(fā)，則它們相應(yīng)系數(shù)有以下關(guān)系：，顯然它們在任意點(diǎn)的函數(shù)值也相同，即從函數(shù)論觀點(diǎn)出發(fā)。反之，若從函數(shù)論觀點(diǎn)出發(fā)，則，這時(shí)域中所有元素都是的根。但是是一個(gè)次數(shù)不超過的多項(xiàng)式，在中至多有個(gè)根，而前述有無限多個(gè)根，這個(gè)矛盾證明必有，

6、即從代數(shù)觀點(diǎn)有。五、簡述題1、答：定義：設(shè)是一個(gè)整環(huán)，如果中每一個(gè)不等于的非單位元素均可寫成：，其中是不可約元素，并且如果還有，其中也是不可約元素，則必有，且適當(dāng)調(diào)整的順序后，有，則稱是因式分解惟一環(huán)。2、答：定理：任何實(shí)系數(shù)次多項(xiàng)式至少有一個(gè)復(fù)數(shù)根。高等代數(shù)專題研究作業(yè)4一、單項(xiàng)選擇題：1-5：bdcaa二、填空題1、 2、 3、4、， 5、三、計(jì)算題1、解：把輛小轎車視為一輛，與輛大卡車排隊(duì)有種方法，而小轎車又有種停放方法，所以一共有種停放方法。2、解：用相同元素的重復(fù)排列公式：，不同的擺法有：種。3、解：展開合并同類項(xiàng)后共有：展開后每一項(xiàng)都是5次多項(xiàng)式，它的不同項(xiàng)實(shí)際上是從6個(gè)元素中取5

7、個(gè)元素的方法數(shù)項(xiàng)，而的系數(shù)為：從3元素中取2個(gè)a，2個(gè)b，1個(gè)c，即為，所以的系數(shù)為30。4、解：設(shè)表示能被整除而不大于2000的自然數(shù)集合，這時(shí)，根據(jù)定理：5、解：用遞推公式：，對(duì)樓梯作歸納：當(dāng)，只有一個(gè)臺(tái)階，只有一種走法；當(dāng)，可以一步一階，也可一步兩階，；當(dāng)，可一步一階，也可一步兩階一階或一階兩階，；，。6、解：設(shè)，1）至少參加一項(xiàng)比賽的人數(shù)：2）只參加四百米比賽的人數(shù)：。四、證明題1、證明：因，所以，所以。令，上式化為，所以，當(dāng)時(shí)，有。2、證明：因，兩邊對(duì)求導(dǎo)一次，得到，上式兩邊乘以，再對(duì)求導(dǎo)，得，令，整理得。五、簡述題答：抽屜原理：如果把件東西裝入個(gè)抽屜，則至少有1個(gè)抽屜里的東西不少于

8、2件。抽屜原理：設(shè)都是正整數(shù)；和進(jìn)有件東西放進(jìn)個(gè)抽屜，則第1個(gè)抽屜至少有件東西，或第2個(gè)抽屜至少有件東西，或第個(gè)抽屜至少有件東西，其中至少有一條必成立。if we dont 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 fas celebrations - said he wanted to meet i

9、vory 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 alway

10、s 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

11、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 fifas anti-racism taskforce, is in london for the football associations 150th anniversary celebrations and will attend c

12、itys 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 its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened disciplinary proceedings against cskafor

13、 the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body, has also ordered an immediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in a statement the russian side added: we

14、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 precipitat

15、ion 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 cruisin

16、g 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

17、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, hell 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 int

18、o 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 parachu