数学专业英语课文句子翻译ok

There are many other less direct benefits the students of geometry may gain……mathematicinans to culture and

civilization

学习几何的学生可以获得许多其他不太直接的利益，

这些人当中必须包括训练在英语语言的精确使用和分

析一个新情况或者问题直达要害的能力，以及利用毅力，创造性和逻辑思维解决是问题。欣赏大自然的创作将是几何研究的副产品。学生还应该发展数学和数学家们对我们的文化和文明作出的贡献的认知。

10-A Although dependence and independence are properties of sets of elements, we also apply these terms to the elements themselves. For example, the elements in an independent set are called independent elements. 虽然相关和无关是元素集的属性，我们也适用于这些元素本身。例如，在一个独立设定的元素被称为独立元素。If s is finite set, the foregoing definition agrees with that given in Chapter 8 for the space nV. However, the present definition is not restricted to finite sets. 如果S是有限集，同意上述定义与第8章中给出的空间nV，然而，目前的定义不局限于有限集。If a subset T of a set S is dependent, then S itself is dependent. This is logically equivalent to the statement that every subset of an independent set is independent. 如果集合S的子集T是相关的，然后S本身是相关的，这在逻辑上相当于每一个独立设置的子集是独立的语句。If one element in S is a scalar multiple of another, then S is dependent. 如果S中的一个元素是另一个集中的多个标量的，则S是相关的。If S 0,then S is dependent. 若S 0，则S 是相关的。The empty set is independent. 空集是无关的。Many examples of dependent and independent sets of vectors in V were discussed in Chapter 8. The following examples illustrate these concepts in function spaces. In each case the underlying linear space V is the set of all real-valued function defined on the real line. V中的向量的相关和无关设置的许多例子是在第8章讨论。下面的例子说明这些概念在函数空间。在每个基本情况下，线性空间V是实线定义的所有实值函数集。Let

1)(),(sin)(,cos)(32221 tuttuttu for all real t. The Pythagorean identity show

that0321 uuu, so the three functions 321,,uuu are dependent. 321,,uuu是相关的。Let kkttu )(for k=0,1,2,…, and t real. The set ,...},,{210uuuS is independent. To prove this, it suffices to show that for each n the n+1 polynomials nuuu,...,,10 are independent. A relation of the form

0kk uc me a ns that

(10.1)

0k k t c for all real t. When t=0, this gives 00 c . Differentiating (10.1) and setting t=0, we find that 01 c. Repeating the process, we find that each coefficient kc is zero. If naa,...,1 are distinct real numbers, the n exponential functions xanxan exuexu )(,...,)(1 1 are independent. We can prove this by induction on n. The result holds trivially when n=1. Therefore, assume it is true for n-1 exponential functions and consider scalars ncc,...,1 such that (10.2)

n kx ak ke c 1 0 Let Ma be the largest of the n numbers naa, (1)

Multiplying both members of (10.2) by xaM e , we obtain (10.3)

n kx aak Mke c 1 )(0 If Mk , th

e

number Mkaa is negative. Therefore, when x in Equation(10.3), each term with Mk tends to zero and we find that 0 Mc. Deleting the Mth term from (10.2) and applying the induction hypothesis, we find that each of the remaining n-1 coefficients kc is zero. Let S be an independent set consisting of k elements in a linear space V and let L(S) be the subspace spanned by S. Then every set of k+1 elements in L(S) is dependent. 设S是一个独立的由k个元素组成的线性空间V，L（S）是S的子空间.每隔K +1的元素在子空间L（S）是相关的。Proof. When nVV ,Theo rem 10.5 reduces to Theorem 8.8.If we examine the proof of Theorem 8.8, we find that it is based only on the fact that nV is a linear space and not on any other special property of nV. Therefore the proof

given for Theorem 8.8 is valid for any linear space V. 证明。当nVV ，定理10.5降低到8.8定理。如果我们研究证明定理8.8，我们发现，这是唯一的事实是一个线性空间上没有任何其他特殊财产。因此，定理8.8的证明有效期为任何线性空间V。

11-A predicates

Statements involving variables, such as “x>3”, ”x+y=3”, ”x+y=z” are often found in mathematical assertion and in computer programs. These statements are neither true nor false when the values of the variables are not specified. In this section we will discuss the ways that propositions can be produced from such statements. 包含变量的语句，比如“x>3”, ”x+y=3”, ”x+y=z”常出现在数学论断中和计算机程序中，若未给语句中的所有变量赋值，则不能判定该语句是真是假，本节要讨论由这种语句生成命题的方法。The statement “x is greater than 3”has two parts. The first part, the variables, is the subject of the statement. The second part-the predicate, “is greater than 3”-refers to a property that the subject of the statement can have. 语句“x大于3”分成两部分，第一部分，变量，是语句的主语。第二部分，谓语，“大于3”，指的是语句主语具有的性质。We can denote the statement “x is greater t han 3” by P(x), where P denote the predicate “is greater than 3” and x is the variable. The statement P(x) is also said to be the value of the propositional function P at x. once a value has been assigned to the variable x, the statements P(x) becomes a proposition and has a truth value. 把语句“x大于3”记为P(x), 其中P表示谓词“大于3”，而x是变量。语句P(x)也称为命题函数P在x点处的值。一旦赋予x一个值，语句P(x)就成为一个命题,有了真值。