中山大学~线性代数期末总复习.ppt

1、REVIEW FOR THE FINAL EXAM,Gao ChengYing Sun Yat-Sen University Spring 2007,Linear Algebra and Its Application,REVIEW FOR THE FINAL EXAM,Chapter 1 Linear Equations in Linear Algebra Chapter 2 Matrix Algebra Chapter 3 Determinants Chapter 4 Vector Spaces Chapter 5 Eigenvalues and Eigenvectors Chapter

2、6 Orthogonality and Least Squares Chapter 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 Solut

3、ion 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 a

4、ugmented 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, or 2. Exactly one solution, or 3. Infinitely many solutions.,consistent,inconsistent,1.1 Systems of Linear Equations,Solving

5、 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 li

6、near system which has unknown variable,1.1 Systems of Linear Equations,Existence and Uniqueness Questions Two fundamental questions about a linear system 1. 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 uniq

7、ue?,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 echelo

8、n 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 sub

9、matrix 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 genera

10、l solution of the following linear system Solution:,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 augmen

11、ted 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 vec

12、tors 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;

13、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 equation Ax = 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 Matri

14、x 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

15、 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

16、 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 Ax Example . Compute Ax, where Solution.,1.4 The Matrix Equation Ax=b,4. Properties of the Matrix-V

17、ector Product Ax Theorem 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 so

18、lution (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 redu

19、ced 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 So

20、lution 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 -

21、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

22、： 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 a

23、re 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

24、 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 Vec

25、tors 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

26、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, the

27、n the set is linearly dependent.,Example,Determine by inspection if the given set is linearly dependent Solution a. The set contains 4 vectors, each has 3 entries. Dependent b. The zero vector is in the set Dependent c. Neither is a multiple of the other Independent,1.8 Linear Transformations,Linear

28、 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 Tra

29、nsformation,Theorem 10 Let T: Rn Rm be a linear transformation. Then there exists a unique matrix A such that T(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 Line

30、ar Transformation,Example: Find the standard matrix A for the dilation transformation T(x) = 3x, for x in R2 Solution:,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 Questions

31、 Definition (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 Transform

32、ation,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 spa

33、n 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 c

34、orresponding entries in A and B Scalar 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 Multiplic

35、ation,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 Multiplicati

36、on,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 (

37、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.

38、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 p

39、ositive 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 Inve

40、rse 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 ma

41、trix, 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

42、-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 Matr

43、ix,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

44、.2 The Inverse of a Matrix,Solution: Since A I, we conclude that A is invertible by Theorem 7 and To 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

45、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

46、xAx is one-to-one. g. The equation Ax=b has at least one solution for each b in Rn h. 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 Ma

47、trices,1. Partitions of Matrices 2. Addition and Scalar Multiplication 3. Multiplication of Partitioned Matrices 4. Inverses of Partitioned Matrices,2.4 Partitioned Matrices,1. Partitions of Matrices Example : The matrix 2 3 partitioned matrix where,2.4 Partitioned Matrices,2. Addition and Scalar Mu

48、ltiplication 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 Matri

49、ces,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 Factorization 2. An LU Factorization Algorithm,2.5 Matrix Factorizations,LU Factorization A is mn matrix and can be row reduced to echelon f

50、orm 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 of Solut

51、ion ： Since A has four rows, L should be 44,2.5 Matrix Factorizations,Row reduction of A to an echelon form U:,2.5 Matrix Factorizations,At each pivot column, divide the highlighted entries by the pivot and place the result into L:,Chapter 3 Determinants, 3.1 Introduction to Determinants 3.2 Propert

52、ies of Determinants 3.3 Cramers Rule, Volume, and Linear Transformations,3.1 Introduction to Determinants,Definition：,Given A = aij, the (i, j)-cofactor of A is the number Cij given by This formula is called a cofactor expansion across the first row of A.,3.1 Introduction to Determinants,Theorem 1 T

53、he determinant of an nn matrix A can be computed by a cofactor expansion across any row or down any column. The expansion across the ith row using the cofactors in (4) is The cofactor expansion down the jth column is,3.1 Introduction to Determinants,Example： Compute det A, where Solution： Compute,3.

54、1 Introduction to Determinants,Theorem 2 If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A.,3.2 Properties of Determinants,1. Properties of Determinants,Theorem 3 Row Operations Let A be square matrix. a. If a multiple of one row of A is added to another

55、 row to produce a matrix B, then det B = det A. b. If two rows of A are interchanged to produce B, then det B = -det A. c. If one row of A is multiplied by k to produce B, then det B = kdet A.,3.2 Properties of Determinants,Example：Compute det A, where Solution Compute An interchange of rows 2 and 3

56、 reverses the sign of the determinant, so,3.2 Properties of Determinants,Theorem 4 A square matrix A is invertible if and only if det A 0.,Theorem 5 If A is an nn matrix, then det AT = det A.,Chapter 4 Vector Spaces, 4.1 Vector Spaces and Subspaces 4.2 Null Spaces, Column Spaces, and Linear Transfor

57、mation 4.3 Linearly Independent Sets; Bases 4.4 Coordinate System 4.5 The Dimension of a Vector Space 4.6 Rank,4.1 Vector Spaces and Subspaces,1. Vector Spaces Definition A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplic

58、ation by scalars (real numbers), subject to the ten axioms below. The axioms must hold for all u, v and w in V and for all scalars c and d. (1) u + v is in V. (2) u + v =v + u. (3) (u + v) + w = u + (v + w). (4) There is a zero vector 0 in V such that u + 0 = u. (5) For each u in V, there is vector

59、u in V such that u + (- u) = 0. (6) cu is in V. (7) c(u + v) = cu + cv. (8) (c + d)u = cu + du. (9) c(du) = (cd)u. (10) 1u = u.,4.1 Vector Spaces and Subspaces,2. Subspaces Definition A subspace of a vector space V is a subset H of V that has three properties: a. The zero vector of V is in H2. b. H is closed under vector addition. That is, for each u