数学专业英语2-10翻译

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 n V . 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n V ，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===t u t t u t t u for all real t. The Pythagorean identity show that 0321=-+u u u , so the three functions 321,,u u u are dependent.

321,,u u u 是相关的。

Let k k t t u =)(for k=0,1,2,…, and t real. The set ,...},,{210u u u S = is independent. To prove this, it suffices to show that for each n the n+1 polynomials n u u u ,...,,10 are independent. A relation of the form ∑=0k k

u c

means 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 k c is zero.

If n a a ,...,1 are distinct real numbers, the n exponential functions x a n x a n

e x u e x u ==)(,...,)(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 n c c ,...,1 such that

(10.2)

∑==n

k x

a k

k e

c

1

Let M a be the largest of the n numbers n a a ,...,1. Multiplying both members of (10.2) by x a M

e -, we obtain

(10.3)

∑=-=n

k x

a a k

M k e

c

1

)(0

If M k ≠, the number M k a a - is negative. Therefore, when +∞→x in Equation(10.3), each term with M k ≠ tends to zero and we find that 0=M c . Deleting the Mth term from (10.2) and applying the induction hypothesis, we find that each of the remaining n-1 coefficients k c 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 n V V =,Theorem 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 n V is a linear space and not on any other special property of n V . Therefore the proof

given for Theorem 8.8 is valid for any linear space V.

证明。当n V V ，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 。