中山大学-线性代数期末总复习ppt课件.ppt

《中山大学-线性代数期末总复习ppt课件.ppt》由会员分享，可在线阅读，更多相关《中山大学-线性代数期末总复习ppt课件.ppt（137页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

1、 REVIEW FOR THE FINAL EXAMGao ChengYingSun Yat-Sen UniversitySpring 2007Linear Algebra and Its ApplicationREVIEW FOR THE FINAL EXAMChapter 1 Linear Equations in Linear AlgebraChapter 2 Matrix AlgebraChapter 3 Determinants Chapter 4 Vector SpacesChapter 5 Eigenvalues and EigenvectorsChapter 6 Orthogo

2、nality and Least SquaresChapter 7 Symmetric Matrices and Quadratic FormsCHAPTER 1Linear Equations in Linear AlgebraChapter 1 Linear Equation in Linear Algebra 1.1 Systems of Linear Equations 1.2 Row Reduction and Echelon Forms 1.3 Vector Equation 1.4 The Matrix Equation Ax = b 1.5 Solution Sets of L

3、inear Systems 1.7 Linear Independence 1.8 Introduction to Linear Transformation 1.9 The Matrix of a Linear Transformation1.1 Systems of Linear Equationsp1. linear equation a1x1 + a2x2+ . . . + anxn = b Systems of Linear Equationsmnmnmmnnnnbxaxaxabxaxaxabxaxaxa221122222121112121111.1 Systems of Linea

4、r EquationspConfficient matrix and augmented matrixCoefficient matrixaugmented matrixmnmmnnaaaaaaaaa212222111211mmnmmnnbaaabaaabaaa212222211112111.1 Systems of Linear EquationspA solution to a system of equations nA system of linear equations has either 1. No solution, or2. Exactly one solution, or3

5、. Infinitely many solutions.consistentinconsistent1.1 Systems of Linear EquationspSolving a Linear SystemnElementary Row Operations 1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows. 3. (Scaling) Multiply all entries in a row b

6、y a nonzero constant.Examplesp 1. Solving a Linear Systemp 2. Discuss the solution of a linear system which has unknown variable1.1 Systems of Linear EquationspExistence and Uniqueness QuestionsnTwo fundamental questions about a linear system1. Is the system consistent; that is, does at least one so

7、lution exist?2. If a solution exists, is it the only one; that is, is the solution unique?1.2 Row Reduction and Echelon FormspThe following matrices are in echelon form:pThe following matrices are in reduced echelon form:pivot position1.2 Row Reduction and Echelon FormspTheorem 1 Uniqueness of the R

8、educed Echelon Form1.2 Row Reduction and Echelon FormspThe Row Reduction Algorithm Step1 Begin with the leftmost nonzero column. Step2 Select a nonzero entry in the pivot column as a pivot. Step3 Use row replacement operations to create zeros in all positions below the pivot. Step4 Apply steps 1-3 t

9、o the submatrix that remains. Repeat the process until there are no more nonzero rows to modify. Step5 Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot.1.2 Row Reduction and Echelon FormspSolution of Linear Systems (Using Row Reduction) eg. Find th

10、e general solution of the following linear systemSolution:1.2 Row Reduction and Echelon Forms The associated system now is The general solution is:1.2 Row Reduction and Echelon FormspTheorem 2 Existence and Uniqueness Theorem1.3 Vector EquationspAlgebraic Properties of For all u, v, w in and all sca

11、lars c and d: where u denotes (-1)u1.3 Vector EquationspSubset of - Span v1,vp is collection of all vectors that can be written in the form with c1,cp scalars.1.4 The Matrix Equation Ax = bp1.Definition If A is an mn matrix, with column a1,an, and if x is in Rn, then the product of A and x, denoted

12、by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is:1.4 The Matrix Equation Ax = bpTheorem 3 If A is an mn matrix, with column a1,an, and if b is in Rm, the matrix equationAx = b has the same solution set as the vector equation which, in turn

13、, has the same solution set as the system of linear equation whose augmented matrix is 1.4 The Matrix Equation Ax = bp2. Existence of Solutions The equation Ax=b has a solution if and only if b is a linear combination of columns of A.Example. Is the equation Ax=b consistent for all possible b1,b2,b3

14、? 1.4 The Matrix Equation Ax=bpSolution Row reduce the augmented matrix for Ax=b:p The equation Ax=b is not consistent for every b.= 0 (for some choices of b)1.4 The Matrix Equation Ax=bpTheorem 4 Let A be an mn matrix. Then the following statements are logically equivalent. That is, for a particula

15、r A, either they are all true statements or they are all false. a. For each b in Rm, the equation Ax = b has a solution. b. Each b in Rm is a linear combination of the columns of A. c. The columns of A span Rm. d. A has a pivot position in every row. 1.4 The Matrix Equation Ax=bp3. Computation of Ax

16、Example . Compute Ax, whereSolution.1.4 The Matrix Equation Ax=bp4. Properties of the Matrix-Vector Product AxTheorem 5 If A is an mn matrix, u and v are vectors in Rn, and c is a scalar, then:1.5 Solution Set of Linear Systemsp1. Solution of Homogeneous Linear Systems p2. Solution of Nonhomogeneous

17、 Systems1.5 Solution Set of Linear Systemsp1. Homogeneous Linear Systems Ax = 0 - trivial solution (平凡解) - nontrivial solution (非平凡解)pThe homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.1.5 Solution Set of Linear SystemspExample Solve

18、the Homogeneous Linear Systems pSolution (1) Row reductionExamplep(2) Row reduction to reduced echelon formp(3) The general solution1.5 Solution Set of Linear Systemsp2. Solution of Nonhomogeneous Systems eg. Describe all solutions of Ax=b, whereSolution 1.5 Solution Set of Linear SystemsThe general

19、 solution of Ax=b has the formThe solution set of Ax=b in parametric vector form1.5 Solution Set of Linear SystemspTheorem 6 Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w= p + vh, where vh is a

20、ny solution of the homogeneous equation Ax=0.1.7 Linear Independencep1. Definition - Linear Independence An indexed set of vectors v1,vp in Rn is said to be linearly independent if the vector equation has only the trivial solution. - Linear Dependence The set v1,vp is said to be linearly dependent i

21、f there exist weights c1,cp , not all zero, such that1.7 Linear IndependencepExample： a. Determine if the set v1,v2,v3 is linearly independent. b. If possible, find a linear dependence relation among v1,v2,v3.Example pa. Row reduce the augmented matrix Clearly, x1 and x2 are basic variables, and x3

22、is free. Each nonzero value of x3 determines a nontrivial solution. Hence v1,v2,v3 are linearly dependent.Example 1pb. completely row reduce the augmented matrix: Thus, x1=2x3, x2=-x3, and x3 is free. Choose x3=5, Then x1=10 and x2=-5. So one possible linear dependence relations among v1,v2,v3 is 1.

23、7 Linear IndependenceThe condition of linear independence: pFor Matrix Columns - if and only if the equation Ax=0 has only the trivial solution.pFor Sets of One or Two Vectors - if and only if neither of the vectors is a multiple of the other.pFor Sets of Two or More Vectors - Theorem 7 (Characteriz

24、ation of Linearly Dependent Sets)p4. Linear Independence of Sets of Two or More Vectors Theorem 7 (Characterization of Linearly Dependent Sets) An indexed set S = v1,vp of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others.

25、 In fact, if S is linearly dependent and v10, then some vj is a linear combination of the preceding vectors, v1,vj-1.b1.7 Linear IndependencepTheorem 8 If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set v1,vp in Rn is linearly d

26、ependent if p n. pTheorem 9 If a set S = v1,vp in Rn contains the zero vector, then the set is linearly dependent.Example pDetermine by inspection if the given set is linearly dependentSolutiona. The set contains 4 vectors, each has 3 entries. Dependentb. The zero vector is in the set Dependentc. Ne

27、ither is a multiple of the other Independent1.8 Linear TransformationspLinear Transformations Definition A transformation T is linear if: (a) T(u+v) = T(u) +T(v) for all u, v in the domain of T; (b) T(cu) =cT(u) for all u and all scalars c. If T is a linear transformation, then T(0) = 0 and for all

28、u, v and scalars c, d: T(cu+dv) = cT(u) +dT(v)1.9 The Matrix of A Linear Transformation pTheorem 10 Let T: Rn Rm be a linear transformation. Then there exists a unique matrix A such thatT(x) =Ax for all x in Rn In fact, A is the mn matrix whose j th column is the vector T(ej), where ej is the j th c

29、olumn of the identity matrix in Rn:A = T(e1) T(en)1.9 The Matrix of A Linear TransformationpExample: Find the standard matrix A for the dilation transformation T(x) = 3x, for x in R2Solution:1.9 The Matrix of A Linear Transformationp2. Geometric Linear Transformations of R2 会求线性变换的标准矩阵，不要求记住1.9 The

30、Matrix of A Linear Transformationp3. Existence and Uniqueness QuestionsDefinition (1) A mapping T : Rn-Rm is said to be onto Rm if each b in Rm is the image of at least one x in Rn (满射)(2) A mapping T : Rn-Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn (单射)1.9 The Ma

31、trix of A Linear Transformation1.9 The Matrix of A Linear TransformationpTheorem 11 Let T: Rn-Rm be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.pTheorem 12 Let T: Rn-Rm be a linear transformation and let A be the standard matrix fo

32、r T. Then: a. T maps Rn onto Rm if and only if the columns of A span Rm b. T is one-to-one if and only if the columns of A are linearly independentChapter 2 Matrix Algebra 2.1 Matrix Operation 2.2 The Inverse of a Matrix 2.3 Characterizations of Invertible Matrices 2.4 Partitioned Matrices 2.5 Matri

33、x Factorizations2.1 Matrix OperationpSum A + B: the sum of the corresponding entries in A and BpScalar Multiples cA : the multiples c of all the entries in A The sum A + B is defined only when A and B are the same size.2.1 Matrix OperationpTheorem 1Let A, B and C be matrices of the same size, and le

34、t r and s be scalars.2.1 Matrix Operationp2. Matrix Multiplication2.1 Matrix Operation If A is an mn matrix, and if B is an np matrix with columns b1,bp, then the product AB is the mp matrix whose columns are Ab1,Abp. That is Each column of AB is a linear combination of the columns of A using weight

35、s from the corresponding column of B.p2. Matrix Multiplication2.1 Matrix OperationpExample : Compute AB, where Solution Write , and compute:2.1 Matrix Operation Row- Column Rule for Computing AB If the product AB is defined, then the entry in row i and column j of AB is the sum of the product of cor

36、responding entries from row i of A and column j of B. If (AB)ij denotes the (i, j)-entry in AB, and if A is an mn matrix, then2.1 Matrix Operationp3. Properties of Matrix Multiplication Theorem 2 Let A be an mn matrix, and let B and C have sizes for which the indicated sums and products are defined.

37、 2.1 Matrix OperationpWarnings: 1. In general, AB BA. 2. The cancellation laws do not hold for matrix multiplication. That is, if AB =AC, then it is not true in general that B = C. 3. If a product AB is the zero matrix, you cannot conclude in general that either A = 0 or B = 0.2.1 Matrix Operationp4

38、. Power of a Matrix If A is an mn matrix and if k is a positive integer, then Ak denotes the product of k copies of A: 2.1 Matrix Operation Theorem 3 Let A and B denote matrices, whose sizes are appropriate for the following sums and products. The transpose of a product of matrices equals the produc

39、t of their transposes in the reverse order.2.2 The Inverse of a Matrixp1. The Inverse of a Matrix For matrix: An nn matrix A is said to be invertible if there is an nn matrix C such that CA = I and AC =I. (I = In) singular matrix: not invertible matrix nonsigular matrix: invertible matrix2.2 The Inv

40、erse of a Matrix Theorem 5 If A is an invertible nn matrix, then for each b in Rn, the equation Ax = b has the unique solution x = A-1b. 2.2 The Inverse of a MatrixpExample : Use the inverse of the matrix to solve the system. Solution : This system is equivalent to Ax = b, so2.2 The Inverse of a Mat

41、rix Theorem 6 a. If A is an invertible matrix, then A-1is invertible and b. If A and B are nn invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, c. If A is an invertible matrix, then so is AT, and the inverse of AT is th

42、e transpose of A-1. That is, 2.2 The Inverse of a Matrixp3. An Algorithm for Finding A-1 Algorithm for Finding A-1 Row reduce the augmented matrix A I. If A is row equivalent to I, then A I is row equivalent to I A-1. Otherwise, A does not have an inverse.2.2 The Inverse of a MatrixpExample : Find t

43、he inverse of the matrix A, if it exists. Solution:2.2 The Inverse of a MatrixSolution: Since A I, we conclude that A is invertible by Theorem 7 andTo check the final answer:2.3 Characterizations of Invertible Matrices Theorem 8 The Invertible Matrix Theorem Let A be a square nn matrix. Then the fol

44、lowing statements are equivalent. That is, for a given A, the statements are either all true or all false.a. A is an invertible matrix. b. A is row equivalent to the nn identity matrix.c. A has n pivot positions. d. The equation Ax=0 has only the trivial solution.e. The columns of A form a linearly

45、independent set.f. The linear transformation xAx is one-to-one.g. The equation Ax=b has at least one solution for each b in Rnh. The columns of A span Rn. i.The linear transformation xAx maps Rn onto Rn.j. There is an nn matrix C such that CA = I.k. There is an nn matrix D such that AD = I. l. AT is

46、 an invertible matrix. 2.4 Partitioned Matricesp1. Partitions of Matricesp2. Addition and Scalar Multiplicationp3. Multiplication of Partitioned Matricesp4. Inverses of Partitioned Matrices2.4 Partitioned Matricesp1. Partitions of MatricespExample : The matrix2 3 partitioned matrixwhere2.4 Partition

47、ed Matricesp2. Addition and Scalar Multiplication A and B : Matrices of same size and partitioned in the same way A + B: the same partition of the ordinary matrix sum A + B. each block is the sum of corresponding blocks of A and B. cA: Multiplication of a partitioned matrix A by a scalar c computed

48、block by block.2.4 Partitioned Matrices Theorem 10 Column-Row Expansion of AB If A is mn matrix and B is np, then3. Multiplication of Partitioned Matrices: AB2.5 Matrix Factorizationsp1. The LU Factorizationp2. An LU Factorization Algorithm2.5 Matrix FactorizationspLU Factorization A is mn matrix an

49、d can be row reduced to echelon form without row interchanges. Then A can be written in: A = LU where L is mm lower triangular matrix and U is mn echelon form of A. Such a factorization is called an LU factorization. L : invertible, a unit lower triangular matrix2.5 Matrix FactorizationspExample ： F

50、ind an LU factorization ofpSolution ： Since A has four rows, L should be 442.5 Matrix FactorizationspRow reduction of A to an echelon form U:2.5 Matrix FactorizationspAt each pivot column, divide the highlighted entries by the pivot and place the result into L:Chapter 3 Determinants 3.1 Introduction