数学专业英语[1]－The Real Number System

hehe123

A differential equation is an equation between specified derivatives of a function, its

valves,and known quantities.Many laws of physics are most simply and naturally formu-

lated as differential equations (or DE’s, as we shall write for short).For this reason,DE’s

3 J1 Q5 h! R8 u( \, I5 P" {

have been studies by the greatest mathematicians and mathematical physicists since the

time of Newton..

Ordinary differential equations are DE’s whose unknowns are functions of a single va-

* g" T9 W- c+ D1 v

riable;they arise most commonly in the study of dynamic systems and electric networks.

# W- O- r1 |* V8 x7 n2 p( c

They are much easier to treat than partial differential equations,whose unknown functions

0 `* {8 ~# A8 a8 e& z+ x1 I) i

depend on two or more independent variables.

Ordinary DE’s are classified according to their order. The order of a DE is defined as

the largest positive integer, n, for which an n-th derivative occurs in the equation. This

! m4 E0 n/ x7 q0 t% j! a

chapter will be restricted to real first order DE’s of the form

* B9 H8 C& l, q# T4 H

Φ(x, y, y′)=0 (1)

Given the function Φof three real variables, the problem is to determine all real functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the following sense.

DEFINITION A solution of (1)is a differentiable function f(x) such that

, w% ^8 T- _3 A4 P3 }9 u* y

Φ(x. f(x),f′(x))=0 for all x in the interval where f(x) is defined.

EXAMPLE 1. In the first-other DE

x+yy′=0 (2)

2 N6 e2 r8 o3 a$ R# V$ B- z, Q

the function Φ is a polynomial function Φ(x, y, z)=x+ yz of three variables in-

volved. The solutions of (2) can be found by considering the identity

+ J/ d! l, d8 n; z* n9 t9 o$ D

d(x²+y²)/d x=2(x+yyˊ).From this identity,one sees that x²+y² is a con-

4 _- o9 r5 c4 u e/ w# `

stant if y=f(x) is any solution of (2).

The equation x²+y²=c defines y implicitly as a two-valued function of x,

) ]% c; c! F+ n1 g: V: X: D/ b

for any positive constant c.Solving for y,we get two solutions,the(single-valued)

# m5 k. Y" O% N

functions y=±(c-x²)^0.5 ,for each positive constant c.The graphs of these so-

lutions,the so-called solution curves,form two families of scmicircles,which fill the upper half-plane y>0 and the lower half-plane y>0,respectively.

+ L/ c; b( ^% `

On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solutions

which cross the x-axis,except possibly at the origin.This fact is easily overlooked,

because the solution curves appear to cross the x-axis;hence yˊdoes not exist,and the DE (2) is not satisfied there.

The preceding difficulty also arises if one tries to solve the DE(2)for yˊ. Dividing through by y,one gets yˊ=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.

DEFINITION. A normal first-order DE is one of the form

yˊ=F(x,y) (3)

3 w' T9 a9 {5 C: l' i' ]4 K

In the normal form yˊ=-x/y of the DE (2),the function F(x,y) is continuous in the upper half-plane y>0 and in the lower half-plane where y<0;it is undefined on the x-axis.

- R3 Q5 ^1 B$ F ; n5 m, L" v7 m9 v4 z

: J g( u; A; d2 P) [3 z

Fundamental Theorem of the Calculus.

- ?& p, F' Q1 c, \+ }. i) r9 ~

6 U2 P$ f) ?: R) H7 j

/ `$ v$ ]' F& c: |" ^

The most familiar class of differential equations consists of the first-order DE’s of the form

yˊ=g(x) (4)

Such DE’s are normal and their solutions are descried by the fundamental thorem of the calculus,which reads as follows.

FUNDAMENTAL THEOREM OF THE CALCULUS. Let the function g(x)in DE(4) be continuous in the interval a<x<b.Given a number c,there is one and only one solution f(x) of the DE(4) in the interval such that f(a)=c. This solution is given by the definite integral

f(x)=c+∫_a^xg(t)dt , c=f(a) (5)

This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a<x<b in the xy-plane.Second,it describes in precise terms the class of functions g(x) considered.And third, it asserts the existence and uniqueness of a solution,given the “initial condition”f(a)=c.

We recall that the definite integral

0 R, B& X& z( M S/ p

∫_a^xg(t)dt=lim(maxΔt_k->0)Σg(t_k)Δt_k , Δt_k=t_k-t_k-1 (5ˊ)

is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral ∫ g(x) dx to give meaning to the definite integral ∫_a^xg(t)dt,provided only that g(t) is continuous.Such functions as the error function crf x =(2/(π)^0.5)∫₀^xe^-t² dt and the sine integral function SI(x)=∫_x^∞[(sin t )/t]dt are indeed commonly defined as definite integrals.

Solutions and Integrals

& w/ a) c, Y$ ]3 P

5 c& q- B$ b) E) F7 j$ n* E

; w# m+ d R6 p# a2 ^6 }

According to the definition given above a solution of a DE is always a function. For example, the solutions of the DE x+yyˊ=0 in Example I are the functions y=± (c-x²)^0.5,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily described by the equation x²+y²=c,describing a family of circles centered at the origin.In what sense can such a family of curves be considered as a solution of the DE ?To answer this question,we require a new notion.

; R5 m8 L( k" i

DEFINITION. An integral of DE(1)is a function of two variables,u(x,y),which assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.

In the above example, the function u(x,y)=x²+y² is an integral of the DE x+yyˊ =0,because,upon replacing the variable y by any function ±( c-x²)^0.5,we obtain u(x,y)=c.

The second-order DE

d²x/dt²=-x (2ˊ)

1 \! m8 P. L. `* v

becomes a first-order DE equivalent to (2) after setting dx/dx=y:

( X j0 [. a5 T

y ( dy/dx )=-x (2)

/ r6 V& R$ {4 v

As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE.When the DE (2ˊ)

is interpreted as equation of motion under Newton’s second law,the integrals

, o/ n1 \+ s% `' k2 ^% X

c=x²+y² represent curves of constant energy c.This illustrates an important principle:an integral of a DE representing some kind of motion is a quantity that remains unchanged through the motion.

1 F: @& Z$ n" i4 f0 g

' C" v! U5 R2 d/ y. q# E

hehe123

Vocabulary , T% p9 U3 t2 F: M1 W' v3 @5 k( @

differential equation 微分方程 error function 误差函数

ordinary differential equation 常微分方程 sine integral function 正弦积分函数

order 阶,序 diameter 直径

derivative 导数 curve 曲线

known quantities 已知量 replace 替代

unknown 未知量 substitute 代入

single variable 单变量 strip 带形

dynamic system 动力系统 exact differential 恰当微分

electric network 电子网络 line integral 线积分

partial differential equation 偏微分方程 path of integral 积分路径

classify 分类 endpoints 端点

polynomial 多项式 general solution 通解

several variables 多变量 parameter 参数

family 族 rigorous 严格的

semicircle 半圆 existence 存在性

half-plane 半平面 initial condition 初始条件

region 区域 uniqueness 唯一性

normal 正规,正常 Riemann sum 犁曼加

identity 恒等(式)

hehe123

Notes

1. The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs in the question.

这是另一种定义句型,请参看附录IV.此外要注意nth derivative 之前用an 不用a .

2. This chapter will be restricted to real first order differential equations of the form

Φ(x,y,yˊ)=0

意思是;文章限于讨论形如 Φ(x,y,yˊ)=0的实一阶微分方程.

有时可以用of the type代替 of the form 的用法.

The equation can be rewritten in the form yˊ=F(x,y).

3. Dividing through by y,one gets yˊ=-x/y,…

划线短语意思是:全式除以y

4. As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE

这里x²+y²=c 因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.

5. Their solutions are described by the fundamental theorem of the calculus,which reads as follows.

意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下.

句中reads as follows 就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.

hehe123

Exercise

Ⅰ.Translate the following passages into Chinese:

1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫_c M(x,y) dx +N(x,y) dy is the same for all paths of integration c in D, which have the same endpoints.

Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.

2. For any normal first order DE yˊ=F(x,y) and any initial x₀ , the initial valve problem consists of finding the solution or solutions of the DE ,for x>x₀ which assumes a given initial valve f(x₀)=c.

3. To show that the initial valve problem is well-set requires proving theorems of existence (there is a solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on the initial value).

Ⅱ. Translate the following sentences into English:

1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有

dч(x)/dx=f (x,ч(x))

2) 两边从x₀到x取定积分得

ч(x)-ч(x₀)=∫_x0^x f(x,ч(x)) dx x₀<x<x₀+h

3) 把y₀=ч(x₀)代入上式, 即有

ч(x)=y₀+∫_x0^x f(x,ч(x)) dx x₀<x<x₀+h

4) 因此 y=ч(x) 是积分方程

y=y₀+∫_x0^x f (x,y) dx

定义于x₀<x<x₀+h 的连续解.

Ⅲ. Translate the following sentences into English:

1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数 f (x, dy/dx) 有连续的偏导数.

2) 引入参数dy/dx=p, 则已给方程变为 y=f (x,p).

3) 在 y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx

4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.

5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为

y=f (x,ч(x,c)).

6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解

x=ψ(p,c)

y=f(ψ(p,c)p)

其中p是参数,c是任意常数.

hehe123

Series are a natural continuation of our study of functions. In the previous chapter we found how

to approximate our elementary functions by polynomials, with a certain error term. Conversely, one can define arbitrary functions by giving a series for them. We shall see how in the sections below.

In practice, very few tests are used to determine convergence of series. Essentially, the comparision test is the most frequent. Furthermore, the most important series are those which converge absolutely. Thus we shall put greater emphasis on these.

* S( V A& B N& g0 T

: V9 n# v& i' {3 U. X

Convergent Series

1 P4 h4 J! |6 e. w" k+ [

Suppose that we are given a sequcnce of numbers

# S. I% ~% I- C% m/ z, {% [. ]

a₁,a₂,a₃…

i.e. we are given a number a_n, for each integer ｎ＞1.We form the sums

" P( U7 G# D9 J7 }7 K6 A

S_n=a₁+a₂+…+a_n

; k: ?9 C1 ~8 D+ U: L+ P. w

It would be meaningless to form an infinite sum

a₁+a₂+a₃+…

because we do not know how to add infinitely many numbers. However, if our sums S_n approach a limit as n becomes large, then we say that the sum of our sequence converges, and we now define its sum to be that limit.

) x1 Y% j/ r7 R' P- ?5 @

The symbols

∑_a=1 ^∞ a_n ^{* \/ |8 }/ Z6 o: G2 s! r}

will be called a series. We shall say that the series converges if the sums approach a limit as n becomes large. Otherwise, we say that it does not converge, or diverges. If the seriers converges, we say that the value of the series is

∑_a=1^∞=lim_a→∞S_n=lim_a→∞(a₁+a₂+…+a_n)

In view of the fact that the limit of a sum is the sum of the limits, and other standard properties of limits, we get:

THEOREM 1. Let{ a_n }and { b_n }(n=1,2,…)

be two sequences and assume that the series

∑_a=1 ^∞ a_n∑_a=1^∞ b_n ^{% Z$ X# J$ Z8 j}

- @1 d$ E, u; c+ T

converge. Then ∑_a=1^∞(a_n + b_n ) also converges, and is equal to the sum of the two series. If c is a number, then

9 k) J; F6 K' }4 `/ B# K# M6 t. C

∑ _a=1^∞c a_n =c∑_a=1 ^∞a_{n # Z6 o+ Q8 ^% Y}

% }/ y: f6 d# H

Finally, if s_n=a₁+a₂+…+a_n and t_n=b₁+b₂+…+b_n then

9 I8 Y7 S, y G

∑_a=1^∞a_n ∑ _a=1^∞b_n=lim_a→∞ s_n t_n

In particular, series can be added term by term. Of course , they cannot be multiplied term by term.

We also observe that a similar theorem holds for the difference of two series.

If a series ∑a_n converges, then the numbers a_n must approach 0 as n becomes large. However, there are examples of sequences {an} for which the series does not converge, and yet lim_a→∞a_n=0

' v+ H7 M, a. @% I2 a' o$ ?3 L

Series with Positive Terms

, N. ?3 Q) Y! P- g8 c! e/ ]

Throughout this section, we shall assume that our numbers a_n are ＞ 0. Then the partial sums

S_n=a₁+a₂+…+a_n

+ ?, G% o( J4 S/ p. k6 m1 U

are increasing, i.e.

+ P" P. g7 S, r9 }- B

s₁＜s₂ ＜s₃＜…＜s_n＜s_n+1＜…

If they are approach a limit at all, they cannot become arbitrarily large. Thus in that case there is a number B such that

# Y/ F s. z* ?

S_n< B

" d2 T: i1 B8 b. S5 r2 y, \

for all n. The collection of numbers {s_n} has therefore a least upper bound ,i.e. there is a smallest number S such that

s_n<S

2 v/ w0 E% O) d" d7 m9 g

for all n. In that case , the partial sums s_n approach S as a limit. In other words, given any positive number ε>0, we have

4 }2 Y, l% ?9 L$ u$ G7 U4 R, K, ?

S –ε< s_n < S

9 B; U9 y5 {0 U- b& v

for all n .sufficiently large. This simply expresses the fact that S is the least of all upper bounds for our collection of numbers s_n. We express this as a theorem.

THEOREM 2. Let{a_n}(n=1,2,…)be a sequence of numbers>0 and let

0 K" }! s" Y; y; i/ Y3 i) g) w

S_n=a₁+a₂+…+a_n

If the sequence of numbers {s_n} is bounded, then it approaches a limit S , which is its least upper bound.

* D4 X6 a5 ^" Z7 \5 P

Theorem 3 gives us a very useful criterion to determine when a series with positive terms converges:

" I6 [& F% `( F1 X$ v1 ?2 d1 h

THEOREM 3. Let∑_a=1^∞a_n and∑_a=1^∞ b_n be two series , with a_n>0 for all n and b_n>0 for all n. Assume that there is a number c such that

a_n< c b_n

for all n, and that∑_a=1^∞b_n converges. Then ∑_a=1^∞ a_n converges, and

2 e0 S' @+ \* {& W

∑_a=1^∞a_n ≤ c∑_a=1^∞b_n

[- N" c( J* t

PROOF. We have

a₁+…+a_n≤cb₁+…+cb_n

0 O1 Y9 E- T) b; V; E, P

=c(b₁+…+b_n)≤ c∑_a=1^∞b_n

This means that c∑_a=1^∞b_n is a bound for the partial sums a₁+…+a_n.The least upper bound of these sums is therefore ≤ c∑_a=1^∞b_n, thereby proving our theorem.

Differentiation and Intergration of Power Series.

If we have a polynomial

a₀+a₁x+…+a_nxⁿ

( m/ ]5 K% D1 n; [! L/ @

with numbers a₀,a₁,…,a_n as coefficients, then we know how to find its derivative. It is a₁+2a₂x+…+na_nx^n–1. We would like to say that the derivative of a series can be taken in the same way, and that the derivative converges whenever the series does.

5 L0 {6 C# y; I v% L

THEOREM 4. Let r be a number >0 and let ∑a_nxⁿ be a series which converges absolutely for ∣x∣<r. Then the series ∑na_nx^n-1 also converges absolutely for∣x∣<r.

9 h" j. ~6 U( m& v' K

A similar result holds for integration, but trivially. Indeed, if we have a series ∑_a=1^∞a_nxⁿ which converges absolutely for ∣x∣<r, then the series

$ D6 k: }0 ^* M' x W3 y

∑_a=1^∞a_n/n+1 xⁿ⁺¹=x∑_a=1^∞a_nxⁿ ∕n+1

4 L) G( S7 Z6 W8 Y* C) }' h. @

has terms whose absolute value is smaller than in the original series.

The preceding result can be expressed by saying that an absolutely convergent series can be integrated and differentiated term by term and and still yields an absolutely convergent power series.

0 v" {. ^/ g$ _/ x, l

It is natural to expect that if

f (x)=∑_a=1^∞a_nxⁿ,

/ z& K5 A/ s/ K8 x5 m

then f is differentiable and its derivative is given by differentiating the series term by term. The next theorem proves this.

% l( p! K U2 G' x: ~

THEOREM 5. Let

f (x)=∑_a=1^∞ a_nxⁿ

) a; I. H% `) A! D* M! o

be a power series, which converges absolutely for∣x∣<r. Then f is differentiable for ∣x∣<r, and

3 N6 I: [9 M0 p* M6 [

f′(x)=∑_a=1^∞na_nx^n-1.

. ^" Q% E$ @0 o' [$ l. H

THEOREM 6. Let f (x)=∑_a=1^∞a_nxⁿ be a power series, which converges absolutely for ∣x∣<r. Then the relation

: \/ ^$ `4 M9 i$ [; Q" m% c

∫f (x)d x=∑_a=1^∞a_nxⁿ⁺¹∕n+1

is valid in the interval ∣x∣<r.

We omit the proofs of theorems 4,5 and 6.

4 V( |. y& Z5 D' P7 \+ g- D& t5 \

0 V) u, j+ o9 A, T$ M! |

hehe123

Vocabulary

sequence 序列 positive term 正项

series 级数 alternate term 交错项

approximate 逼近,近似 partial sum 部分和

elementary functions 初等函数 criterion 判别准则(单数)

section 章节 criteria 判别准则(多数)

convergence 收敛(名词) power series 幂级数

convergent 收敛(形容词) coefficient 系数

absolute convergence 绝对收敛 Cauchy sequence 哥西序列

diverge 发散 radius of convergence 收敛半径

term by term 逐项 M-test M—判别法

hehe123

Notes

1. series一词的单数和复数形式都是同一个字.例如:

One can define arbitrary functions by giving a series for them(单数)

The most important series are those which converge absolutely(复数)

2. In view of the fact that the limit of a sum of the limits, and other standard properties of limits, we get:

Theorem 1…

这是叙述定理的一种方式: 即先将事实说明在前面,再引出定理. 此句用in view of the fact that 说明事实,再用we get 引出定理.

3. We express this as a theorem.

这是当需要证明的事实已再前面作了说明或加以证明后,欲吧已证明的事实总结成定理时,常用倒的一个句子,类似的句子还有(参看附录Ⅲ):

We summarize this as the following theorem; Thus we come to the following theorem等等.

4. The least upper bound of these sums is therefore ≤c∑_a=1^∞b_n, thereby proving our theorem.

最一般的定理证明格式是”给出定理…定理证明…定理证毕”,即thereby proving our theorem;或we have thus proves the theorem或This completes the proof等等作结尾(参看附录Ⅲ).

5. 本课文使用较多插入语.数学上常见的插入语有:conversely; in practice; essentially; in particular; indeed; in other words; in short; generally speaking 等等.插入语通常与句中其它成份没有语法上的关系,一般用逗号与句子隔开,用来表示说话者对句子所表达的意思的态度.插入语可以是一个词,一个短语或者一个句子.

hehe123

Exercise

Ⅰ. Translate the following exercises into Chinese:

1. In exercise 1 through 4,a sequence f (n) is defined by the formula given. In each case, (ⅰ)

Determine whether the sequence (the formulae are omitted).

2. Assume f is a non–negative function defined for all x>1. Use the method

suggested by the proof of the integral test to show that

∑_k=1^n-1f(k)≤∫₁ⁿf(x)d x ≤∑_k=2ⁿf(k)

Take f(x)=log x and deduce the inequalities

c•nⁿ•c^-n< n！<c•nⁿ⁺¹•c^-n

Ⅱ. The proof of theorem 4 is given in English as follows(Read the proof through and try to learn how a theorem is proved, then translate this proof into Chinese ):

Proof of theorem 4 Since we are interested in the absolute convergence. We may assume that a_n>0 for all n. Let 0<x<r, and let c be a number such that x<c<r. Recall that lim_a→∞n^1/n=1.

We may write n a_n xⁿ =a_n(n^1/nx)ⁿ. Then for all n sufficiently large, we conclude that n^1/nx<c. This is because n1/n comes arbitrarily close to x and x<c. Hence for all n sufficiently large, we have na_nxⁿ<a_ncⁿ. We can then compare the series ∑na_nxⁿ with∑a_ncⁿ to conclude that∑na_nxⁿ converges. Since∑na_nx^n-1=1/x∑na_nxⁿ, we have proved theorem 4.

Ⅲ. Recall from what you have learned in Calculus about (ⅰ) Cauchy sequence and (ⅱ) the radius of convergence of a power series.

Now give the definitions of these two terms respectively.

Ⅳ. Translate the following sentences into Chinese:

1. 一旦我们能证明,幂级数∑a_nzⁿ 在点z=z₁收敛,则容易证明,对每一z₁∣z∣<∣z₁∣ ,级数绝对收敛;

2. 因为∑a_nzⁿ在z=z₁收敛,于是,由weierstrass的M—判别法可立即得到∑a_nzⁿ在点z,∣z∣<z₁的绝对收敛性;

3. 我们知道有限项和中各项可以重新安排而不影响和的值,但对于无穷级数,上述结论却不总是真的

hehe123

For the definition that follows we assume that we are given a particular field K. The scalars to be used are to be elements of K.

DEFINITION. A vector space is a set V of elements called vectors satisfying the following axioms.

(A) To every pair, x and y ,of vectors in V corresponds a vector x+y,called the sum of x and y, in such a way that.

(1) addition is commutative, x + y = y + x.

6 `4 o! b' E: n9 Q! c

(2) addition is associative, x + ( y + z ) = ( x + y ) + z.

(3) there exists in V a unique vector 0 (called the origin ) such that x + 0 = x for every vector x , and

' g1 m6 p [8 N% d; |' S. _

(4) to every vector x in V there corresponds a unique vector - x such that x + ( - x ) = 0.

(B) To every pair,αand x , where α is a scalar and x is a vector in V ,there corresponds a vector αx in V , called the product of α and x , in such a way that

(1) multiplication by scalars is associative,α(βx ) = (αβ) x

4 _1 L* S5 q; P4 ^) }: G$ l

(2) 1 x = x for every vector x.

% |. L' p, w/ v, [1 A" q9 X

(C) (1) multiplication by scalars is distributive with respect to vector addition,α( x + y ) = αx+βy , and

, u9 k9 E" ~! w

(2)multiplication by vectors is distributive with respect to scalar addition,(α+β) x = αx + βx .

0 j& [& e- x3 @1 F: }0 A

The relation between a vector space V and the underlying field K is usually described by saying that V is a vector space over K . The associated field of scalars is usually either the real numbers R or the complex numbers C . If V is linear space and M真包含于V , and if α u -v belong to M for every u and v in M and every α∈ K , then M is linear subspace of V . If U = { u 1,u 2,…} is a collection of points in a linear space V , then the (linear) span of the set U is the set of all points o the form ∑ c _i u _i , where c _i∈ K ,and all but a finite number of the scalars c_i are 0.The span of U is always a linear subspace of V.

A key concept in linear algebra is independence. A finite set { u ₁,u ₂,…, u _k } is said to be linearly independent in V if the only way to write 0 = ∑ c _i u _i is by choosing all the c _i = 0 . An infinite set is linearly independent if every finite set is independent . If a set is not independent, it is linearly dependent, and in this case, some point in the set can be written as a linear combination of other points in the set. A basis for a linear space M is an independent set that spans M . A space M is finite-dimensional if it can be spanned by a finite set; it can then be shown that every spanning set contains a basis, and every basis for M has the same number of points in it. This common number is called the dimension of M .

. E- j" u- O( B

Another key concept is that of linear transformation. If V and W are linear spaces with the same scalar field K , a mapping L from V into W is called linear if L (u + v ) = L( u ) + L ( v ) and L ( αu ) = α L ( u ) for every u and v in V and α in K . With any I , are associated two special linear spaces:

: f! g' U/ r# g

ker ( L ) = null space of L = L^-1 (0)

= { all x ∈ V such that L ( X ) = 0 }

( }! S8 P" c3 }8 k; w; P% q

Im ( L ) = image of L = L( V ) = { all L( x ) for x∈ V }.

1 d1 c2 L) ~/ Q! R8 }

Then r = dimension of Im ( L ) is called the rank of L. If W also has dimension n, then the following useful criterion results: L is 1-to-1 if and only if L is onto.In particular, if L is a linear map of V into itself, and the only solution of L( x ) = 0 is 0, then L IS onto and is therefore an isomorphism of V onto V , and has an inverse L ^-1 . Such a transformation V is also said to be nonsingular.

Suppose now that L is a linear transformation from V into W where dim ( V ) = n and dim ( W ) = m . Choose a basis {υ₁ ,υ_{2 ,}…,υ_n} for V and a basis {w ₁ ,w₂ ,…,w _m} for W . Then these define isomorphisms of V onto Kⁿ and W onto K^m , respectively, and these in turn induce a linear transformation A between these. Any linear transformation ( such as A ) between Kⁿ and K^m is described by means of a matrix ( a_ij ), according to the formula A ( x ) = y , where x = { x₁ , x ₂,…, x_n } y = { y₁ , y ₂,…, y _m} and

Y _j =Σⁿ_j=i a_ij x_i I=1,2,…,m.

+ i) S$ _4 I; J3 ^ J9 w l

The matrix A is said to represent the transformation L and to be the representation induced by the particular basis chosen for V and W .

If S and T are linear transformations of V into itself, so is the compositic transformation ST . If we choose a basis in V , and use this to obtain matrix representations for these, with A representing S and B representing T , then ST must have a matrix representation C . This is defined to be the product AB of the matrixes A and B , and leads to the standard formula for matrix multiplication.

5 D l1 ^% J; h( D" ^; L4 h

The least satisfactory aspect of linear algebra is still the theory of determinants even though this is the most ancient portion of the theory, dating back to Leibniz if not to early China. One standard approach to determinants is to regard an n -by- n matrix as an ordered array of vectors( u ₁ , u ₂ ,…, u _n ) and then its determinant det ( A ) as a function F( u ₁ , u ₂ ,…, u _n ) of these n vectors which obeys certain rules.

7 D: M) n9 r) @

The determinant of such an array A turns out to be a convenient criterion for characterizing the nonsingularity of the associated linear transformation, since det ( A ) = F ( u ₁ , u ₂ ,…, u _n ) = 0 if and only if the set of vectors u_i are linearly dependent. There are many other useful and elegant properties of determinants, most of which will be found in any classic book on linear algebra. Thus, det ( AB ) = det ( A ) det ( B ), and det ( A ) = det ( A') ,where A' is the transpose of A , obtained by the formula A' =( a _ji ), thereby rotating the array about the main diagonal. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.

Another useful concept is that of eigenvalue. A scalar is said to be an eigenvalue for a transformation T if there is a nonzero vector υ with T (υ) λυ . It is then clear that the eigenvalues will be those numbers λ∈ K such that T -λ I is a singular transformation. Any vector in the null space of T -λ I is called an eigenvector of T associated with eigenvalue λ, and their span the eigenspace, E _λ. It is invariant under the action of T , meaning that T carries E_λ into itself. The eigenvalues of T are then exactly the set of roots of the polynomial p(λ) =det ( T -λ I ).If A is a matrix representing T ,then one has p (λ) det ( A -λI ), which permits one to find the eigenvalues of T easily if the dimension of V is not too large, or if the matrix A is simple enough. The eigenvalues and eigenspaces of T provide a means by which the nature and structure of the linear transformation T can be examined in detail.

hehe123

Vocabulary

linear algebra 线性代数 non-singular 非奇异

field 域 isomorphism 同构

vector 向量 isomorphic 同构

scalar 纯量,无向量 matrix 矩阵(单数)

vector space 向量空间 matrices 矩阵(多数)

span 生成,长成 determinant 行列式

independence 无关(性),独立(性) array 阵列

dependence 有关(性) diagonal 对角线

linear combination 线性组合 triangular 三角形的

basis 基(单数) entry 表值,元素

basis 基(多数) eigenvalue 特征值,本征值

dimension 维 eigenvector 特征向量

linear transformation 线性变换 invariant 不变,不变量

null space 零空间 row 行

rank 秩 column 列

singular 奇异 system of equations 方程组

homogeneous 齐次