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2.1数学、方程与比例
词组翻译
1.数学分支branches of mathematics，算数arithmetics，几何 学geometry，代数学
algebra，三角学trigonometry，高等数学highe r mathematics，初等数学elementary
mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论
function theory，微分方程differential equation
2.命题propositi on，公理axiom，公设postulate，定义definition，定理theorem，引
理lemma，推论deduction
3.形form，数number，数字numeral，数值numerical value，图形 figure，公式
formula，符号notation(symbol)，记法记号sign，图 表chart
4.概念conception，相等equality，成立真true，不成立不真 untrue，等式
equation，恒等式identity，条件等式equation of condition，项术语term，集set，
函数function，常数constant，方 程equation，线性方程linear equation，二次方程
quadratic equation
5.运算operation，加法addition，减法subtractio n，乘法multiplication，除法
division，证明proof，推理deduct ion，逻辑推理logical deduction
6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated
operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric
place

汉译英
（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研
究等活动。
Mathematics comes from man’s social practice, for example, industrial and agricultural
production, commercial activities, military operations and scientific and technological
researches.
（2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。
No modern scientific and technological branches could be regularly developed without
the application of mathematics.
（3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。
Notations are a special and powerful tool of mathematics and are used to express
conceptions and propositions very often.
（4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常
数。
Before 17th century, man confined himself to the elementary mathematics, i. e. ,
geometry, trigonometry and algebra, in which only the constants were considered.
（5）方程与算数的等式不同在于它含有可以参加运算的未知量。
Equation is different from arithmetic identity in that it contains unknown quantity which
can join operations.
（6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。
Equipment is called an equation of condition in that it is true only for certain values of
unknown quantities in it.
（7）方程很有用，可以用它来解决许多实际应用问题。
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Equations are of very great use. We can use equations in many mathematical problems.
（8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。
To solve the equation means to move and change the terms about without making the
equation untrue, until the root of the equation is obtained, which is the value of unknown
term.

英译汉
a has evolved from the operations and rules of study of
arithmetic begins with addition,multiplication,subtraction,and division of
numbers:4+7,37×682,49-22,40÷8.
In algebra we introduce symbols or letters—such as a,b,c,d,x,y,z—to denote
arbitrary numbers and，instead of special cases，we often consider general statements：
a+b,cd,x-y,x÷a.
代数是从算术的 运算和规则当中逐渐发展起来的，算术的研究是从数的加减
乘除开始的。例如4+7，37×682，4 9-22，40÷8。
在代数学里,我们采用符号或字母。例如a,b,c,d,x,y,z来表示任 意的数字，而
不考虑那些特殊情况。我们经常考虑的是一般的表达式，例如a+b,cd,x-y,x÷ a。
language of algebra serves a twofold ,we may use it as a shorthand
to abbreviate and simplify long or complicated ,it proves a convenient
means of generalizing many specific statements.
代数的语言有两个作用。第一个是使用它作为一种速记法去缩减和减化那些又 长又
复杂的表达。第二，它被证明是一种概括许多具体的表达方式的便捷途径。
expressions involve two or more ng symbols tell us which
operation is to be done common grouping symbols are
parentheses,(),brackets.[],and the fraction bar,—.For example,in the expression
2(3+4),we do the addition first and then we do the multiplication:2(3+4)=2(7)=14.
许多数学表达式包含两个 或更多的运算。分组符号告诉我们哪一个运算先做。常见
的分组符号是圆括号，方括号和分数线。例如， 在数学表达公式2（3+4）里。我
们先做加法再做乘法2（3+4）=2（7）=14


2.2 几何与三角
词组翻译
1.学会institution，建筑师 architect, 机械师 machinist, 制图员draftsman, 测量者
surveyor, 木匠carpenter
2.点point, 端点endpoint, 线line, 直线straight line, 线段 line segment, 曲线curved
line, 折线 broken line, 射线ray , 平面 plane,曲面 curved surface
3.立体solid, 柱体cylinder, 立方体cube,球 sphere, 棱锥pyramid,圆锥 cone ,
4.圆circle,圆心 center, 直径diameter, 半径radius, 半圆semicircle, 弦chord, 弧arc,
优弧major arc, 劣弧minor arc
5.角angle, 边side, 三角形triangle, 直角三角形right triangle,斜边 hypotenuse, 直
角边right-angle side
6.长度length,宽度 breadthwidth,厚度 thickness, 位置position
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7.几何的geometrical,立体的 three- dimensional , 弯曲的curved,等距离的
equidistant ,无限的 infinite
8.培养创造力train originality,必须的毅力 necessary perseverance ,
提高鉴赏力raiseimprove the appreciation ability
9.消失了的边界vanishing boundarieslandmarks,有序性和优美感 orderliness and
sense of beauty, 几何图形大量存在geometric forms abound in , 定理成立的先决
条件a prerequisite to a theorem

汉译英
（1）许多专家都认为数学是学习其他科学技术的必备基础和先决条件。
Many experts recognize that mathematics is the necessary foundation and prerequisite of
studying other science technology.
（2）西方国家的专家认为几何起源于巴比伦和埃及人的土地测量技术，其实中国
古代的数学 家对几何做了许多出色的研究。
The western experts think that geometry had its origin in the measurements by the
Babylonians and Egyptians of their lands. Infect, the ancient Chinese mathematicians
made much remarkable study for geometry.
（3）几何的学习使学生在思考问题时更周密和审慎，他们将不会盲目接受任何结
论。
In studying geometry, the student is taught to think clearly and critically and he is led
away from the practice of blind acceptance of any conclusions.
（4）数学培养学生的分析问题的能力，使他们能应用毅力、创造性和逻辑推理来
解决问题。
Studying mathematics can develop the students’ ability to analyze problems and utilizing
perseverance, originality, and logical reasoning in solving the problem.
（5）几何主要不是研究数，而是形，例如三角形，平行四边形和圆，虽然它也与
数有关。
Geometry mainly studies hot numbers but figures such as triangles, parallelograms and
circles, though it is related with numbers.
（6）一个立体（图形）有长、宽和高；面（曲面或平面）有长和宽，但没有厚
度；线（直线 或曲线）有长度，但既没有宽度，也没有厚度；点只有位置，却没
有大小。
A solid (figure) has length, width and height. A surface (curved surface or plane surface)
has length and width, but no thickness. A line (straight line or curved line) has length, but
no width and thickness. A point has position, but no dimension.
（7）射线从某个点出发无限延伸；两条从同一点出发的射线构成了角。这两条射
线称为这个 角的两边，当这两边位于同一直线上且方向相反时，所得的角是平
角。
A ray starts from a point and extends infinitely far. Two rays starting from one point form
an angle, which are called two edges of the angle. When two edges lie in the same line
and have opposite direction named plane angle.
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（8）平面上的闭曲线当其中每一点到一个固定点的距离均相等时叫做圆。这个固
定点称为圆 心，经过圆心且其两个端点在圆周上的线段称为这个圆的直径，直径
的一半叫做半径，这条 曲线的长度叫做周长。
A circle is a closed curve lying in one plane, all points of which are equidistant from a
fixed point. The fixed point called the center. A diameter of a circle is a line segment
through the center of the circle with endpoints on the circle. Half of the diameter is called
radius. The length of the circle is called circumference.


英译汉 geometry an angle is defined as the set of points determined by two
rays l
1
and l
2
having the same endpoint O. 在几何学里从同一点O出发 引出的两
条射线l
1
和l
2
所组成的点的集合叫做角。
trigonometry we often interpret angles as rotations of obtain an angle we
may start with a fixed ray l
1
having endpoint O,and rotate it about O,in a plane,to a
position specified by ray l
2
.We call l
1
the initial side, l
2
the terminal side,and O the
vertex of angle. 在三角学里，我们经常解释角就是射线的旋转。在平面 上，我
们许会从端点是O的射线l
1
开始让它绕着端点O旋转，转到一个位置，由射< br>线l
2
标注。我们把l
1
叫做角的始边，l
2
叫做角 的终边，O叫做角的顶点。
3.A right angle is a 90

angle . An angle

is acute if 0

<

<90

or obtuse if
90

<

<180

.A straight angle is a 180

angle .Two acute angles are complementary
if their sum is 90

.Two positive angles are supplementary if their sum is 180

. 直
角就是一个90
的角。如果0

<

<90

把它叫做锐角，如果90

<

<180

叫做
钝角。平角就是一个180

的角。如果两个锐角的和是90

，那么这两个角互为
余角。如果 两个正角的和是180

，那么这两个角是互为补角。
2.3集合论的基本概念
单词、词组
1.1 集set，子集subset，真子集proper subset，全集universal subset，空集void
empty set，基地集the underlying set
1.2 正数positive number，偶数even integer，图形diagram，文氏图Venn diagram，
哑标dummy index，大括号brace
1.3 可以被整除的be divisible by，两两不同的distinct from each other，确定的
definite，无关紧要的irrelevantinessential
1.4 一样的结论the same conclusion，等同的效果equivalent effect，用大括号表示集
sets are designated by braces，把这个图形记作A：this diagram is designated by
letter A，区别对象to distinguish between objects，证明定理to prove theorems，
把结论可视化to visualize conclusionsconsequences

汉译英
（1）由小于 10 且能被 3 整除的正整数组成的集是整数集的子集。
The set consisting of those positive integers less than 10 which are divisible by 3 is a
subset of the set of all integers.
（2）如果方便，我们通过在括号中列举元素的办法来表示集。
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When convenient, we shall designate sets by displaying the elements in braces.
（3）用符号

表示集的包含关系，也就是说，式子 A

B 表示 A 包含于
B。
The relation

is referred to as set inclusion; A

B means that A is contained in B.
（4）命题 A

B 并不排除 B

A 的可能性。
The statement A

B does not rule out the possibility that B

A.
（5）基础集可根据使用场合不同而改变。
The underlying set may vary from one application to another according to using
occasions.
（6）为了避免逻辑上的困难，我们必须把元素 x 与仅含有元素 x 的集{x}区别开
来。
To avoid logical difficulties, we must distinguish between the element x and the set {x}
whose only element is x.
（7）图解法有助于将集合之间的关系形象化。
Diagrams often help using visualize relationship between sets.
（8）定理的证明仅仅依赖于概念和已知的结论，而不依赖于图形。
The proofs of theorems rely only on the definitions of the concepts and known result, not
on the diagrams.

英译汉
A is the set of all the letters of the alphabet，then listing each of elements would be
tedious. So we write A={a,b,c,…,z}.
如果A是所有字母的集合，那么把每一个其中的字母列举出来将是很冗 长乏味
的，因此我们写出A=｛a，b，c，…,z｝。
the set A，the last element is z. Many sets do not have last elements . Two important
sets are N , the set of natural numbers , and W , the set of whose numbers . To list all the
elements in these sets would be impossible because they go on forever . So we use three
dots and write N={1,2,3,…},W={0,1,2,3,…}.
在集合A里,最后一个元素是z，许多集合 没有最后一个元素，两个重要的集合是
N，自然数集合，和W，整数的集合。把这两个集合里所有的元素 列举出来是不可
能的，因为它们是永远持续下去的，所以我们用三个点来表示，集合N写成
N= {1,2,3,…}，集合W写成W={0,1,2,3，…}。
whole numbers have many important subsets . A whole number is said to be even
if it is divisible by 2；2,6,and 18 are examples of even numbers. A whole number is said
to be odd if it is not divisible by 2 1,7,and 13 are examples of odd numbers . The natural
numbers greater than 1 are called prime or composite , A number is prime if it is divisible
only by 1 and itself , A number is composite if it is divisible by a natural number other
than 1 and itself.
整数 有许多重要的子集。如果一个整数能被2除开就是偶数；2，6，18就是偶数
的例子。一个整数如果不 能被2整除就是奇数；1，7，13就是奇数的例子。大于1
的自然数叫做素数或者合数，如果一个自然 数只能被1和它本身整除，那么这个数
就是素数（质数），如果一个自然数除了能被1和它本身整除外， 还可以被其他的
自然数整除，就叫做合数。
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2.4整数、有理数与实数
1.单词
1）Integer rational number irrational number，real number，negative number，
the negative，real line，real axis，scale，to the leftright of
2）sum，differen ce，product，quotient，power，inequality
3）axiom，the field axiom，the order axiom
4）ordered，entirely complete，Euclidean，appropri ate，distinguished，illuminating
5）can be deduced formula，formula interchangeably，using a set of formulas，
corresponding to an object，proof by induction，the two set to be distinguished

2、汉译英
（1）严格说，这样描述整数是不完整的，因为我们并没有说明“依此类推”或“反复
加 1” 的含义是什么。
Strictly speaking, this description of the positive integers is not entirely complete because
we have not explained in detail what we mean by the expressions “and so on”, or
“repeated addition of 1”.
（2）两个整数的和、差或积是一个整数，但是两个整数的商未必是一个整数。
The sum, difference, or product of two integers is an integer, but the quotient of two
integers need not be an integer.
（3） 这种用几何来表示实数的办法对于帮助我们更好地发现与理解实数的性质是
非常有价值 的。
This device for representing real numbers geometrically is a very worthwhile aid that
helps us to discover and understand better certain properties of real numbers.
（4）几何经常为一些特定的定理提供证明思路（建议） ，而且，有时几何的论证
比纯分析的 （完全依赖于实数公理的）证明更清晰。
The geometry often suggests the method of proof of a particular theorem, and sometimes
a geometric argument is more illuminating than a purely analytic proof (one depending
entirely on the axioms for the real numbers).
（5）一个由实数组成的集若满足如下条件则称为开区间（open interval） 。
If a set consisting of real numbers satisfies the following conditions we call it an open
interval.
（6）实数 a 是-a 的相反数，它们的绝对值相等，且当 a ≠ 0 时，其符号不同。
The real number a is the negative number of –a and their absolute values are equal. When
a ≠ 0, their notations are different.
（7）每个实数刚好对应着实轴上的一点，反之，对实轴上的每一点，有且只有一
个实数与之 对应。
Each real number corresponds to exactly one point on this line and, conversely, each
point on the line corresponds to one and only one real number.
（8）在几何上，实数之间的次序关系可以在数轴上清楚地表示出来。
In geometry, the ordering relation among the real numbers can be expressed clearly in
real axis.

3. 英译汉
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1）一个常见的错误是认为x 是一个负数。但x 可以为正数、 0 或负数，这取决
于x 的值
2）我们在最后一节中介绍的每个属性是唯一的操作，例如，ab =ba和 0 + a= a ，
我们现在考虑一个连接加法和乘法的属性。它被称为分配律或者乘法对加法的分
配，用下面的 公式表所示： a（b + c) = ab + ac；（b + c) =ba + ca。
3） 考虑小数按自然数顺序写出为：1234567891„由于自然数数字不会
中止或重复，这是一个无限 不循环小数，不能转换为两个整数的比的小数叫做无理
数，这组数字是指符号H，H={×：x是无限不 循环小数}


2.5 笛卡儿几何学的基本概念
1. 翻译单词、词组、短语
(1)解析几何analytic geometry 笛卡儿几何Cartesian geometry
三维的three-dimensional, 坐标coordinate,坐标系 coordinate system, 坐标原点 the
origin, 横坐标 abscissa,坐标轴coordinate axis,纵坐标ordinate，象限quadrant,有序
对ordered pair, 尺度scale,单位长度 the unit distance
(2)向量vector, 线段line segment, 垂直的perpendicular,水平的 horizontal, 竖直
的vertical, 相交intersect，交点a point
(3)三角形triangle, 直角三角形 right triangle, 斜边 hypotenuse, 直角边leg,区域
arearegion, 多边形的polygonal, 多边形区域polygonal region,抛物线的parabolic,
抛物线弓形 parabolic segment circular, 圆的circular，圆域circular region
(4)积分的计算integral calculation, 整数的性质integral quality, 微积分的基本定
理 basic theorem of calculusp
(5)对符号做适当认定an appropriate regard for signs,
把一个问题转化为另一个问题to reduce a question to another question,
把条件翻译成表达式to translate these conditions into expressions ，
紧密融合在一起inntimately intertwined，
刻画了该曲线的特征to characterize the curve in question

2.汉译英
（1）计算图形的面积是积分的一种重要应用。
The calculation of figure area is the important application of the integral.
（2）在 x-轴上 O 点右边选定一个适当的点，并把它到 O 点的距离称为单位长
度。
On the x-axis a convenient point is chosen to the right of O and its distance from O is
called the unit distance.
（3）对 xy- 平面上的每一个点都指定了一个数对，称为它的坐标。
Each point in the xy-plane is assigned a pair of numbers, called its coordinates.
（4）选取两条互相垂直的直线，其中一条是水平的，另一条是竖立的，把它们的
交点记作 O， 称为原点。
Two perpendicular reference lines are chosen, one horizontal, the other vertical. Their
point of intersection, denoted by O, is called the origin.
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（5）当我们用一对数（a, b）来表示平面的点时，商定要把横坐标写在第一个位置
上。
When we write a pair of numbers such as (a, b) to represent a point, we agree that the
abscissa or x-coordinate, a, is written first.
（6）微积分与解析几何在它们的发展史上已经互相融合在一起了。
Throughout their historical development, calculus and analytic geometry have been
intimately intertwined.
（7）如果想拓展微积分的范围与应用，需要进一步研究解析几何，而这种研究需
用到向量的 方法。
A deeper study of analytic geometry is needed to extend the scope and applications of
calculus, and this study will be carried out using vector methods.
（8）今后我们要对三维解析几何做详细研究，但目前只限于考虑平面解析几何。
We shall discuss three-dimensional Cartesian geometry in more detail later on; for the
present we confine our attention to plane analytic geometry.


2.6函数的概念与函数思想
1.1）function，domain，range，the identity function，the absolute-value function，
the real- valued
2）cube，volume，edge- length，prime，totality
3)Hooke's law，stretch，d isplacement，spring，constant，proportional
4）schematic representation，plot，image，output，input
5）it is not difficult to imagine，the idea was much too limited

2.汉译英
（1）常用英语字母和希腊字母来表示函数。
Letters of the English and Greek alphabets are often used to denote functions.
（2）若 f 是一个给定的函数，x 是定义域里的一个元素，那么记号 f(x)用来表示
由 f 确定的 对应于 x 的值。
If f is a given function and if x is an object of its domain, the notation f(x) is used to
designate that object in the range which is associated to x by the function f.
（3）该射线将两个坐标轴的夹角分成两个相等的角。
The ray makes equal angles with the coordinates axes.
（4）可以用许多方式给出函数思想的图解说明。
The function idea may be illustrated schematically in many ways.
（5）容易证明，绝对值函数满足三角不等式。
It is easy to proof that the absolute-value function satisfies the triangle inequality.
（6）对于实数 x>0，函数 g(x)表示不超过 x 的素数的个数。
For a given real number x>0, the function g(x) is defined by the number of primes less
than or equal to x.
（7）函数是一种对应，它未必可以表示成一个简单的代数公式。
A function is a correspondence. It is not necessary to be expressed by a simple algebraic
formula.
（8）在函数的定义中，关于定义域和值域中的对象，没对其性质做出任何限制。
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The function idea places no restriction on the nature of the objects in the domain X and in
the range Y.

2.7 序列及其极限 序列及其极限
（1）序列各项对 n 的相关性常利用下标来表示，写成如下形式： a n , x n 等。
The dependence of every team of sequence on n is denoted by using subscript, and we
write a n , x n and so on.
（2）以正整数集为定义域的函数称为序列。
A function whose domain is the set of all positive integers is called an infinite sequence.
（3）一个复值序列收敛当且仅当它的实部和虚部分别收敛。
A complex- valued sequence converges if and only if both the real part and the imaginary
part converge separately.
（4） 一个序列{ a n }若满足： 对任意正数 ε ， 存在另一个正数 N （N可能与
ε 有关） 使得 a n - L < ε 对所有 n ≥ N 成立，就称{ a n }收敛于 L。
A sequence { a n } is said to have a limit L if, for every positive number ε , there is
another positive number N (which may depend on ε ) such that In this case, we say the
sequence { a n } converges to L. an ? L < ε for all n ≥ N.
（5） 重要的是， 该集的每一个成员都用一个正整数标上记号。 这样一来， 就
可以谈论第一项、 第二项和一般项，即第 n 项。
The important thing is that each member of the set has been labeled with an integer so
that we may speak of the first term, the second term and in general, the nth term.
（6）若无另加申明，本章研究的序列都假定具有实的项或复的项。
Unless otherwise specified, all sequences in this chapter are assumed to have real or
complex terms.
（7）作为日常用语，sequence 和 series 是同义词；但作为数学术语，它们表示不
同的概念。
In everyday usage of the English language, the words “sequence” and “series” are
synonyms, but in mathematics these words have special technical meanings.
（8）术语“收敛序列”指的是具有有限极限的序列，因此，极限为无限的序列不是
收敛的， 而是发散的。
The phrase “convergent sequence” is used only for a sequence whose limit is finite. A
sequence with an infinite limit is said to diverge not convergence.

2.8 函数的导数和它的几何意义
（1）差商表示函数 f 在连接 x 与 x+h 的区间上的平均变化率。
The different quotient is referred to as the average rate of the change of f in the interval
joining x to x+h.
（2）速度等于位置函数的导数。
Velocity is equal to the derivative of positing.
（3）由定义导数的过程所提供的几何解释以一种自然的方式导出了关于曲线的切
线思想。
The procedure used to define the derivative has a geometric interpretation which leads in
a natural way to the idea of a tangent line to a curve.
（4）差商表示直线 PQ 与水平线的夹角的正切。
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The difference quotient represents the trigonometric tangent of the angle that PQ makes
with the horizontal.
（5）在直线运动中，速度的一阶导数称为加速度。
For rectilinear motion, the first derivative of velocity is called acceleration.
（6）我们约定 f（0）=f，即函数 f 的零阶导数就等于它本身。
We make the convention that f（0）=f, that is the zeroth derivate is the function itself.
（7）在运动的 9 秒钟内，物体的速度由 v (0) = -144 变成了 v (9) =144，也就是
说，速度总共 增加了每秒 288 英尺。
During the 9 seconds of motion the velocity changes from v (0) = -144 to v (9) =144, that
is, the total increase in velocity is 288 feet per second.
（8）当 α 从 0 增加到π2 时，tan α 所对应的直线趋于竖直位置。As α increases
from 0 to π2 , tan α approach a vertical position.

2.9 微分方程简介
（1）此时，微分方程就有无穷多个解，C的每个值对应一个解。
The differential equation has infinitely many solutions, one for each value of C.
（2）微分方程的阶指的是方程中最高阶导数的阶。
By the order of an equation is meant the order of the highest derivative which appears.
（3）我们可以由已知的粒子运动速度或者加速度计算出粒子的位置。
We could try to compute the position of a moving particle from a knowledge of its
velocity or acceleration.
（4）如果一个微分方程的未知函数是多元函数，则称为偏微分方程。
Ordinary and partial, depend on whether the unknown is a function of just one variable or
of two or more variables.
（5）微分方程的研究直接受到力学、天文学和数学物理的推动。
The study of differential equations has been directly inspired by mechanics, astronomy,
and mathematical physics.
（6）许多应用问题要求我们从方程的解集中选出一个在某个点具有指定值的解。
In many problems it is necessary to select from the collection of all solutions one having
a prescribed value at some point.
（7）确定满足边界条件的解的问题称为边值问题。
The problem of determining such a solution that satisfies boundary condition is called a
boundary-value problem.
（8）人们设计许多高速运行的计算机来对各种积分做出近似估计。
Automatic high-speed computing machines are often designed with this kind of problem
in mind.

2.10 线性空间中的相关与无关集
（1）该式的两边同时关于t积分，我们就得到一个所需要的结论。
Integrating both sides of this formula with respect to t. we can obtain a conclusion we
need.
（2）不难看出，这个命题仅仅建立在该空间是线性的这一事实上，与空间的其他
性质无关。
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We clearly find that this proposition is based only on the fact that this space is a linear
space and not on any other special property of this space.
（3）如果空间不存在有限基，就称该空间是无限维的。
A space is called infinite dimensional if it doesn’t have a finite basis.
（4）假定这个结论对n-1个指数函数成立，我们将证明此结论对n个指数函数也
成立。
Assuming the conclusion is true for n-1 exponential functions, we will prove that it is true
for n exponential function.
（5）这两个定义在逻辑上是互相等价的。
These two definitions are logically equivalence.
（6）设X是线性空间V中k个元素组成的 一个线性无关集合，L（X）是由X张
成的子空间。那么，L（X）的每一个元素都可以表示成X的元素 的线性组合。
Let X be an independent set consisting of k elements in a linear space V and let L（X）
be the subspace spanned by X, then each element of L(X) can be expressed as a linear
combination of element of X.
（7）设V是一个n维线性空间，考虑它的一个基，其元素按给定的次序排列为
，，…，。
Let V be a linear space of dimension n and consider a basis whose elements , , … ,
are take in a given order.
（8）该线性表示的系数构成一个n元组，它由向量x唯一确定。
The coefficients in this linear representation determine an n-tuple of numbers that is
uniquely determined by x.


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