数学专业英语课文句子翻译ok.docVIP

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 means that (10.1) ??0k k t c for all real t. When t=0, this