中山大学线性代数期末总复习ppt课件.ppt

REVIEW FOR THE FINAL EXAM,Gao ChengYingSun Yat-Sen UniversitySpring 2007,Linear Algebra and Its Application,REVIEW FOR THE FINAL EXAM,Chapter 1 Linear Equations in Linear AlgebraChapter 2 Matrix AlgebraChapter 3 Determinants Chapter 4 Vector SpacesChapter 5 Eigenvalues and EigenvectorsChapter 6 Orthogonality and Least SquaresChapter 7 Symmetric Matrices and Quadratic Forms,CHAPTER 1Linear Equations in Linear Algebra,Chapter 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 Linear Systems 1.7 Linear Independence 1.8 Introduction to Linear Transformation 1.9 The Matrix of a Linear Transformation,1.1 Systems of Linear Equations,1. linear equation a1x1 + a2x2+ . . . + anxn = b Systems of Linear Equations,1.1 Systems of Linear Equations,Confficient matrix and augmented matrix,Coefficient matrix,augmented matrix,1.1 Systems of Linear Equations,A solution to a system of equations,A system of linear equations has either 1. No solution, or2. Exactly one solution, or3. Infinitely many solutions.,consistent,inconsistent,1.1 Systems of Linear Equations,Solving a Linear System,Elementary 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 by a nonzero constant.,Examples,1. Solving a Linear System,2. Discuss the solution of a linear system which has unknown variable,1.1 Systems of Linear Equations,Existence and Uniqueness QuestionsTwo fundamental questions about a linear system1. Is the system consistent; that is, does at least one solution exist?2. If a solution exists, is it the only one; that is, is the solution unique?,1.2 Row Reduction and Echelon Forms,The following matrices are in echelon form:The following matrices are in reduced echelon form:,pivot position,1.2 Row Reduction and Echelon Forms,Theorem 1 Uniqueness of the Reduced Echelon Form Each matrix is row equivalent to one and only one reduced echelon matrix.,1.2 Row Reduction and Echelon Forms,The 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 to 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 Forms,Solution of Linear Systems (Using Row Reduction) eg. Find the 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 Forms,Theorem 2 Existence and Uniqueness Theorem A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column that is , if and only if an echelon form of the augmented matrix has no row of the form,1.3 Vector Equations,Algebraic Properties of For all u, v, w in and all scalars c and d: where u denotes (-1)u,1.3 Vector Equations,Subset 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 = b,1.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 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 = b,Theorem 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, has the same solution set as the system of linear equation whose augmented matrix is,1.4 The Matrix Equation Ax = b,2. 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?,1.4 The Matrix Equation Ax=b,Solution Row reduce the augmented matrix for Ax=b: The equation Ax=b is not consistent for every b.,=, 0 (for some choices of b),1.4 The Matrix Equation Ax=b,Theorem 4 Let A be an mn matrix. Then the following statements are logically equivalent. That is, for a particular 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=b,3. Computation of AxExample . Compute Ax, whereSolution.,1.4 The Matrix Equation Ax=b,4. 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 Systems,1. Solution of Homogeneous Linear Systems 2. Solution of Nonhomogeneous Systems,1.5 Solution Set of Linear Systems,1. Homogeneous Linear Systems Ax = 0 - trivial solution (平凡解) - nontrivial solution (非平凡解)The 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 Systems,Example Solve the Homogeneous Linear Systems Solution (1) Row reduction,Example,(2) Row reduction to reduced echelon form(3) The general solution,1.5 Solution Set of Linear Systems,2. Solution of Nonhomogeneous Systems eg. Describe all solutions of Ax=b, where,Solution,1.5 Solution Set of Linear Systems,The general solution of Ax=b has the form,The solution set of Ax=b in parametric vector form,1.5 Solution Set of Linear Systems,Theorem 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 any solution of the homogeneous equation Ax=0.,1.7 Linear Independence,1. 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 if there exist weights c1,cp , not all zero, such that,1.7 Linear Independence,Example： a. Determine if the set v1,v2,v3 is linearly independent. b. If possible, find a linear dependence relation among v1,v2,v3.,Example,a. Row reduce the augmented matrix,Clearly, x1 and x2 are basic variables, and x3 is free. Each nonzero value of x3 determines a nontrivial solution. Hence v1,v2,v3 are linearly dependent.,Example 1,b. 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,1.7 Linear Independence,The condition of linear independence: For Matrix Columns - if and only if the equation Ax=0 has only the trivial solution.For Sets of One or Two Vectors - if and only if neither of the vectors is a multiple of the other.For Sets of Two or More Vectors - Theorem 7 (Characterization of Linearly Dependent Sets),4. 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. In fact, if S is linearly dependent and v10, then some vj is a linear combination of the preceding vectors, v1,vj-1.,b,1.7 Linear Independence,Theorem 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 dependent if p n. Theorem 9 If a set S = v1,vp in Rn contains the zero vector, then the set is linearly dependent.,Example,Determine 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. Neither is a multiple of the other Independent,1.8 Linear Transformations,Linear 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 u, v and scalars c, d: T(cu+dv) = cT(u) +dT(v),1.9 The Matrix of A Linear Transformation,Theorem 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 column of the identity matrix in Rn:A = T(e1) T(en),1.9 The Matrix of A Linear Transformation,Example: Find the standard matrix A for the dilation transformation T(x) = 3x, for x in R2Solution:,1.9 The Matrix of A Linear Transformation,2. Geometric Linear Transformations of R2 会求线性变换的标准矩阵，不要求记住,1.9 The Matrix of A Linear Transformation,3. 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 Matrix of A Linear Transformation,1.9 The Matrix of A Linear Transformation,Theorem 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.Theorem 12 Let T: Rn-Rm be a linear transformation and let A be the standard matrix for 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 independent,Chapter 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 Matrix Factorizations,2.1 Matrix Operation,Sum A + B: the sum of the corresponding entries in A and BScalar 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 Operation,Theorem 1,Let A, B and C be matrices of the same size, and let r and s be scalars.,2.1 Matrix Operation,2. Matrix Multiplication,2.1 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 weights from the corresponding column of B.,2. Matrix Multiplication,2.1 Matrix Operation,Example : 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 corresponding 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, then,2.1 Matrix Operation,3. 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.,2.1 Matrix Operation,Warnings: 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 Operation,4. 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 product of their transposes in the reverse order.,2.2 The Inverse of a Matrix,1. 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 matrix,2.2 The Inverse 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 Matrix,Example : Use the inverse of the matrix to solve the system. Solution : This system is equivalent to Ax = b, so,2.2 The Inverse of a Matrix,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 the transpose of A-1. That is,2.2 The Inverse of a Matrix,3. 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 Matrix,Example : Find the inverse of the matrix A, if it exists. Solution:,2.2 The Inverse of a Matrix,Solution: 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 following 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 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 an invertible matrix.,2.4 Partitioned Matrices,1. Partitions of Matrices2. Addition and Scalar Multiplication3. Multiplication of Partitioned Matrices4. Inverses of Partitioned Matrices,2.4 Partitioned Matrices,1. Partitions of MatricesExample : The matrix2 3 partitioned matrixwhere,2.4 Partitioned Matrices,2. 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 block by block.,2.4 Partitioned Matrices,Theorem 10 Column-Row Expansion of AB If A is mn matrix and B is np, then,3. Multiplication of Partitioned Matrices: AB,2.5 Matrix Factorizations,1. The LU Factorization2. An LU Factorization Algorithm,2.5 Matrix Factorizations,LU Factorization A is mn matrix and 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 matrix,2.5 Matrix Factorizations,Example ： Find an LU factorization ofSolution ： Since A has four rows, L should be 44,2.5 Mat