第八章有理整数环 8-1有理整数环的基本概念 8.1.1有理整数环的基本概念 全体整数所组成的集合中有两种运算:加法和乘法,而且它们满足下面运算法则: （1)加法满足结合律; （2)加法满足加换律 （3)有一个数0,是对任意整数a,0+a=a; （4)对任意整数a,存在整数b,使b+a=0 （5)乘法满足结合律 （6)有一个数1,是对任意整数a,la=a

1 The Number-Picking Game Here is a game that you and I could play that reveals a strange property of expectation. 3, First, you think of a probability density function on the natural numbers. Your distri- bution can be absolutely anything you like. For example, you might choose a uniform distribution on 1, 2, ... 6, like the outcome of a fair die roll. Or you might choose a bi- probability, provided that,...,n. You can even give every natural number a non-zero nomial distribution on 0, 1 he sum of all probabilities is 1

1 Equivalence Relations 【Definition】A relation R is defined on a set S if for every pair of elements (a, b), a, b S, a R b is either true or false. If a R b is true, then we say that a is related to b

一、单项选择题 题分 20 题号 １ ２ ３ ４ ５ ６ ７ ８ ９ １０ 得分 答案 D B B A A C C C A A 题号 １１ １２ １３ １４ １５ １６ １７ １８ １９ ２０ 答案 C C C C C A A C A B

一、单项选择题 题分 20 题号 １ ２ ３ ４ ５ ６ ７ ８ ９ １０ 得分 答案 B A A B C D C A D C 题号 １１ １２ １３ １４ １５ １６ １７ １８ １９ ２０ 答案 C A A B B C D A B D

第四章4-3线性映射与线性变换 4.3.1线性映射的定义 定义设U,V为数域K上的线性空间,φ:U→V为映射,且满足以下两个条件: i)、(a+)=(a)+(),(a,B∈U); i)、(ka)=k(a),(a∈U,k∈K), 则称为(由U到V的)线性映射, 由数域K上的线性空间U到V的K的线性映射的全体记为Hom(U,V),或简记为 Hom(U,). 定义中的i和)二条件可用下述一条代替 (ka+1)=k(a)+kq(B),(a,B∈U,k,l∈K)

根据哈密尔顿一凯莱定理,任给数域P上一个级矩阵A,总可以找到数域 P上一个多项式f(x),使f(A)=0.如果多项式f(x)使f(A)=0,就称f(x)以A 为根当然,以为A根的多项式是很多的,其中次数最低的首项系数为1的以A为 根的多项式称为A的最小多项式这一节讨论应用最小多项式来判断一个矩阵能 否对角化的问题

定理1(无穷小与函数极限的关系) 在自变量的同一变化过程x→x(或x→∞)中,函数f(x) 具有极限A的充分必要条件是(x)=A+a,其中a是无穷小 简要证明令a=f(x)-A,则fx)-a

Cloning vectors: 克隆载体 to clone a gene in a vector Expression vectors: 表达载体 to express a gene from a vector Integration vectors: 整合载体 to integrate a gene in a genome through a vector Cloning vectors 1 Plasmid vecters 2 Bacteriophage vectors 3 Cosmids & BACs 4 Eukaryotic vectors Cloning vectors: allowing the exogenous DNA to be inserted, stored, and manipulated mainly at DNA level. expression vectors: allowing the exogenous DNA to be inserted, stored, and expressed

1.证明a(t)是常向量的充要条件是a(t)=0 2.证明()x()()x()+i()x 4.设向量函数a(t)满足a(t)a'=0,a(t)xa=0,证明a()是常向量。 5.证明F(t)=(2t-1,t2-2,-t2+4t)为共面向量函数