数学专业英语－Linear Algebra

yunjiang · 发表于 2004-5-6 09:33:33

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> 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.
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> DEFINITION. A vector space is a set V of elements called vectors satisfying the following axioms.
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> (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.
(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
(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
(2) 1 x = x for every vector x.
(C) (1) multiplication by scalars is distributive with respect to vector addition,α( x + y ) = αx+βy , and
(2)multiplication by vectors is distributive with respect to scalar addition,(α+β) x = αx + βx .
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 ci 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 .
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:
 ker ( L ) = null space of L = L-1 (0)
 = { all x ∈ V such that L ( X ) = 0 }
 Im ( L ) = image of L = L( V ) = { all L( x ) for x∈ V }.
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 ,υ2 ,…,υn} for V and a basis {w 1 ,w2 ,…,w m} for W . Then these define isomorphisms of V onto Kn and W onto Km , respectively, and these in turn induce a linear transformation A between these. Any linear transformation ( such as A ) between Kn and Km is described by means of a matrix ( aij ), according to the formula A ( x ) = y , where x = { x1 , x 2,…, xn } y = { y1 , y 2,…, y m} and 
 
 Y j =Σnj=i aij xi I=1,2,…,m.
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.
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.
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 ui 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 Eλ 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.

yunjiang · 发表于 2004-5-6 09:33:49

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Vocabulary<

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0cm 0cm 0pt; TEXT-INDENT: 21pt; mso-layout-grid-align: none">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

yunjiang · 发表于 2004-5-6 09:34:03

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Notes<

0cm 0cm 0pt; tab-stops: 0cm; mso-layout-grid-align: none">1. If U = { u 1 , u 2 ,…}is a collection of points in a linear<

0cm 0cm 0pt; tab-stops: 0cm; mso-layout-grid-align: none">space V , then the (linear) span of the set U is the set of all points of the form ∑ c i u i , wwhere c i∈ K ,and all but a finite number of scalars c I are 0.意思是:如果 U = { u 1 , u 2 ,…}是线性空间 V 的点集,那么集 U 的(线性)生成是所有形如 ∑ c i u i 的点集,这里 c i ∈ K ,且除了有限个 ci 外均为0. 2. A finite set { u 1 , u 2 ,…, u k} is said to be linearly independent if the only way to write 0 = ∑ c i u I is by choosing all the c i= 0.这一句可以用更典型的句子表达如下: A finite set { u 1, u 2 ,…, u k } is said to be linearly independent in V if ∑c i u i is by choosing all the c i = 0.这里independent 是形容词,故用linearly修饰它. 试比较F(x) is a continuous periodic function.这里periodic 是形容词但它前面的词却用continuous 而不用continuously,这是因为continuous 这个词不是修饰periodic而是修饰作为整体的名词periodic function. 3. Then these define isomorphisms of V onto Kn and W onto KM respectively, and these in turn induce a linear transformation A between these.这里第一个these代表前句的两个基(basis);第二个these代表isomorphisms;第三个these代表什么留给读者自己分析. 4. The least satisfactory aspect of linear algebra is still the theory of determinants- 意思是:线性代数最令人不满意的方面仍是有关行列式的理论.least satisfactory 意思是:最令人不满意. 5. 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.意思是:如果方阵是三角形的,即所有在主对角线上方的元素均为零,那末det( A ) 刚好就是对角线元素的乘积.这里meaning that 可用that is to say 代替,turns out to be解为”结果是”.

yunjiang · 发表于 2004-5-6 09:34:15

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Exercise<

0cm 0cm 0pt 36pt; TEXT-INDENT: -36pt; tab-stops: 36.0pt; mso-layout-grid-align: none">I. Answer the following questions:<

0cm 0cm 0pt; mso-layout-grid-align: none"> 1. How can we define the linear independence of an infinite set? 2. Let T be a linear transformation (T: V → W ) whose associated matrix is A.Give a criterion for the non-singularity of the transformation T.3. Where is the entry a45 of a m -by- n matrix( m>4; n>5) located ?4. Let A , B be two rectangular matrices.Under what condition is the product matrix well-defined ? II.Translate the following two examples and their proofs into Chinese: 1.Example1. Let uk= tk ,k=0,1,2,... and t real. Show that the set ｛u 0,u1,u2,…｝ is independent. Proof: By the definition of independence of an infinite set, it suffices to show that for each n ,the n+1 polynomials u0,u1,...,un are independent.A relation of the form ∑nk=0 ckuk=0 means ∑nk=0 ck tk=0 for all t.When t=0,this gives c0 =0.Differentiating both sides of ∑nk=0 ck tk =0 and setting t=0,we find that c1=0.Repeating the process,we find that each cocfficient is zero2. Example 2. Let V be afinite dimensional linear space, Then every finite basis for V has the same number of elements.Proof: Let S and T be two finite bases for V. Suppose S consists of k elemnts and T consists of m elements.Since S is independent and spans V ,every set of k+1 elements in V is dependent.Therefore every set of more than k elements in V is dependent. Since T is an independent set , we must have m<k. The same argument with S and T interchanged shows that k<m. Hence k=m. III.Translate the following sentences into English: 1.设 A 是一矩阵。若其行数 n 等于其列数 m ，则称 A 是一方阵；若n ≠m，则称A 是一矩形阵。 2.设 T 是一线性变换，则 T 的特征值刚好是多项式 P（λ）=det（T- λE ）的根。 3.形如Σnk=1cik xk =ci i=1，2，...,m 的一组方程称为 m 个线性方程，n 个未知数的方程组。若所有 ci=0，则称上述方程组为一齐次方程组。

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