杅悝蚳珛荎逄ㄜLinear Algebra

yunjiang 楷桶衾 2004-5-6 09:33:33

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 setVof elements called vectors satisfying the following axioms.
 (A) To every pair, xandy ,of vectors inVcorresponds a vector x+y,called the sum ofxandy, 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 inVa unique vector 0 (called the origin ) such that x + 0 = x for every vector x , and
(4)to every vector x inVthere corresponds a unique vector - x such that x + ( - x ) = 0.
(B) To every pair,汐and x , where 汐 is a scalar and x is a vector inV,there corresponds a vector 汐x inV, called the product of 汐 and x , in such a way that 
(1)multiplication by scalars is associative,汐(汕x ) = (汐汕) x
(2)1 x = x for every vector x.
(C)(1) multiplication by scalars is distributive with respect to vector addition,汐( x + y ) = 汐x+汕y , and
(2)multiplication by vectors is distributive with respect to scalar addition,(汐+汕) x = 汐x + 汕x .
The relation between a vector spaceVand the underlying fieldKis usually described by saying thatVis a vector space overK . The associated field of scalars is usually either the real numbersR or the complex numbersC . IfV is linear space andM淩婦漪衾V , and if 汐 u -v belong toMfor everyuandvinM and every 汐﹋K, then Mis linear subspace ofV . If U = { u 1,u 2,＃} is a collection of points in a linear spaceV , then the (linear) span of the setUis 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 ofV.
 A key concept in linear algebra is independence. A finite set { u 1,u 2,＃, u k } is said to be linearly independent inV 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 spaceMis an independent set that spansM .A spaceMis 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 forMhas the same number of points in it. This common number is called the dimension ofM.
Another key concept is that of linear transformation. IfVandWare linear spaces with the same scalar fieldK , a mappingLfromVintoWis called linear ifL (u + v ) =L( u ) + L ( v ) andL ( 汐u ) = 汐 L ( u )for everyuandvinVand 汐 inK . With anyI , are associated two special linear spaces:
 ker (L) = null space ofL=L-1(0)
 = { all x ﹋ Vsuch thatL ( X ) = 0 }
 Im ( L ) = image ofL=L( V )= { all L( x ) for x﹋ V }.
Thenr = dimension of Im ( L ) is called the rank ofL. IfW also has dimensionn, then the following useful criterion results: L is 1-to-1 if and only if L is onto.In particular, ifL is a linear map ofV into itself, and the only solution ofL( x ) = 0 is 0, then L IS onto and is therefore an isomorphism of V ontoV , and has an inverseL -1 . Such a transformationVis also said to be nonsingular.
Suppose now thatLis a linear transformation fromVintoWwhere dim ( V ) = n anddim( W ) = m . Choose a basis {耒1 ,耒2 ,＃,耒n} forV and a basis {w 1 ,w2 ,＃,w m} forW . Then these define isomorphisms ofV ontoKnandW ontoKm , respectively, and these in turn induce a linear transformationAbetween these.Any linear transformation ( such asA) betweenKnandKm is described by means of a matrix ( aij ), according to the formulaA ( x )= y , wherex = { x1 , x 2,＃, xn } y = { y1 , y 2,＃, y m} and 
 
 Y j=曳nj=iaij xi I=1,2,＃,m.
The matrix A is said to represent the transformationL and to be the representation induced by the particular basis chosen for V andW .
If S and T are linear transformations ofV into itself, so is the compositic transformationST . If we choose a basis inV , and use this to obtain matrix representations for these, withArepresentingS andB representingT , then ST must have a matrix representationC. This is defined to be the productAB of the matrixesAandB , 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 arrayA turns out to be a convenient criterionfor 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 vectorsui are linearly dependent. There are many other useful and elegant propertiesof 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') ,whereA' is the transpose ofA , obtained by the formulaA' =( 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 transformationT if there is a nonzero vector耒 withT (耒)竹耒 . It is then clear that the eigenvalues will be those numbers 竹﹋K such thatT -竹 I is a singular transformation. Any vector in the null space ofT -竹 I is called an eigenvector ofT associated with eigenvalue 竹, and their span the eigenspace,E 竹.It is invariant under the action ofT , meaning thatT carriesE竹 into itself. The eigenvalues ofTare thenexactly the set of roots of the polynomialp(竹) =det ( T-竹 I ).IfA is a matrix representingT ,then one hasp (竹) det ( A -竹I ), which permits one to find the eigenvalues ofT easily if the dimension ofV is not too large, or if the matrixA is simple enough. The eigenvalues and eigenspaces of T provide a means by which the nature and structure of the linear transformationT can be examined in detail.

yunjiang 楷桶衾 2004-5-6 09:33:49

Vocabulary linear algebra 盄俶測杅 non-singular 準✽祑field 郖 isomorphism 肮凳vector 砃講 isomorphic 肮凳scalar 曾講,拸砃講 matrix 撻淝(等杅)vector space 砃講諾潔 matrices 撻淝(嗣杅)span 汜傖,酗傖 determinant 俴蹈宒independence 拸壽(俶),黃蕾(俶) array 淝蹈dependence 衄壽(俶) diagonal 勤褒盄linear combination 盄俶郪磁 triangular ʊ褒倛腔basis 價(等杅) entry 桶硉,啋匼basis 價(嗣杅) eigenvalue 杻涽硉,掛涽硉dimension峎 eigenvector 杻涽砃講linear transformation 盄俶曹遙 invariant 祥曹,祥曹講null space 錨諾潔 row 俴rank 窏 column蹈singular ✽祑 system of equations 源最郪 homogeneous ぅ棒

yunjiang 楷桶衾 2004-5-6 09:34:03

Notes1. If U = { u 1 , u 2 ,＃}is a collection of points in a linearspaceV , 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.砩佷岆:⺼彆 U= { u 1 , u 2 ,＃}岆盄俶諾潔 V 腔萸摩,饒繫摩U 腔(盄俶)汜傖岆垀衄倛⺼ ﹉ c i u i 腔萸摩,涴爵 c i﹋K ,ブ壺賸衄癹跺 ci俋歙峈0. 2. Afinite 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.涴珨曆褫眕蚚載萎倰腔曆赽桶湛⺼狟: A finite set { u 1, u 2 ,＃, u k } is said to be linearly independent inV if ﹉c i u i is by choosing all the c i = 0.涴爵independent 岆倛⺍棵,嘟蚚linearly党庉坳. 彸掀誕F(x) is a continuous periodic function.涴爵periodic 岆倛⺍棵筍坳ゴ醱腔棵�蚚continuous 奧祥蚚continuously,涴岆秪峈continuous 涴跺棵祥岆党庉periodic奧岆党庉釬峈淕极腔靡棵periodic function. 3. Then these define isomorphisms ofV ontoKnandW ontoKM respectively, and these in turn induce a linear transformationA between these.涴爵菴珨跺these測桶ゴ曆腔謗跺價(basis);菴媼跺these測桶isomorphisms;菴ʊ跺these測桶妦繫隱跤黍氪赻撩煦昴. 4.The least satisfactory aspect of linear algebra is still the theory of determinants- 砩佷岆:盄俶測杅郔鍔�祥雛砩腔源醱゛岆衄壽俴蹈宒腔燴蹦.least satisfactory 砩佷岆:郔鍔�祥雛砩. 5. Ifa 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.砩佷岆:⺼彆源淝岆ʊ褒倛腔,撈垀衄婓翋勤褒盄奻源腔啋匼歙峈錨,饒藺det( A ) 試疑憩岆勤褒盄啋匼腔傚儅.涴爵meaning that 褫蚚that is to say 測杸,turns out to be賤峈§賦彆岆§.

yunjiang 楷桶衾 2004-5-6 09:34:15

ExerciseI. Answer the following questions:1. How can we define the linear independence of an infinite set?2. LetTbe a linear transformation (T: V ↙ W) whose associated matrix is A.Give a criterion for the non-singularity of the transformationT.3. Where is the entry a45 of am -by-nmatrix( m>4; n>5) located ?4. LetA, Bbe 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 eachn,the n+1 polynomials u0,u1,...,un are independent.Arelation 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 thatc1=0.Repeating the process,we find that each cocfficient is zero2.Example 2. LetVbe afinite dimensional linear space, Then every finite basis forV has the same number of elements.Proof: LetSandT be two finite bases for V. SupposeS consists ofk elemnts and T consists of melements.SinceS is independent and spansV,every set ofk+1elements inVis dependent.Therefore every set of more than k elements in Vis dependent. Since Tis an independent set , we must have m<k. The same argument with S and Tinterchanged shows that k<m. Hence k=m. III.Translate the following sentences into English: 1.扢A 岆珨撻淝﹝�［俴杅n 脹衾［蹈杅 mㄛ寀備A 岆珨源淝˙�n≧mㄛ寀備A 岆珨撻倛淝﹝ 2.扢 T 岆珨盄俶曹遙ㄛ寀 T 腔杻涽硉試疑岆嗣砐宒Pㄗ竹ㄘ=detㄗT- 竹E ㄘ腔跦﹝ 3.倛⺼曳nk=1cik xk =cii=1ㄛ2ㄛ...,m腔珨郪源最備峈m跺盄俶源最ㄛn跺帤眭杅腔源最郪﹝�垀衄ci=0ㄛ寀備奻扴源最郪峈珨ぅ棒源最郪﹝

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杅悝蚳珛荎逄ㄜLinear Algebra