中山大学~线性代数期末总复习.ppt

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,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 6 Orthogonality and Least Squares,Chapter 7 Symmetric Matrices and Quadratic Forms,CHAPTER 1,Linear 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,a,1,x,1,+,a,2,x,2,+.+,a,n,x,n,=,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,or,2.Exactly one solution,or,3.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 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 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 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 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 v,1,v,p,is collection of all vectors that can be written in the form,with c,1,c,p,scalars.,1.4 The Matrix Equation Ax=b,1.Definition,If,A,is an,m,n,matrix,with column a,1,a,n,and if,x,is in R,n,then the product of,A,and,x,denoted by,A,x,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,m,n,matrix,with column a,1,a,n,and if,b,is in R,m,the matrix equation,A,x,=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,A,x,=,b,has a solution if and only if,b,is a linear combination of columns of,A,.,Example.,Is the equation,A,x,=,b,consistent for all possible b,1,b,2,b,3,?,1.4 The Matrix Equation Ax=b,Solution,Row reduce the augmented matrix for,A,x,=,b,:,The equation,A,x,=,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,m,n,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 R,m,the equation A,x,=b has a solution.,b.Each b in R,m,is a linear combination of the columns of A.,c.The columns of A span R,m,.,d.A has a pivot position in every row.,1.4 The Matrix Equation Ax=b,3.Computation of,A,x,Example.,Compute Ax,where,Solution.,1.4 The Matrix Equation Ax=b,4.Properties of the Matrix-Vector Product Ax,Theorem 5,If,A,is an,m,n,matrix,u and v are vectors in R,n,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+v,h,where v,h,is any solution of the homogeneous equation Ax=0.,1.7 Linear Independence,1.Definition,-,Linear Independence,An indexed set of vectors v,1,v,p,in R,n,is said to be linearly independent if the vector equation,has only the trivial solution.,-,Linear Dependence,The set v,1,v,p,is said to be linearly dependent if there exist weights c,1,c,p,not all zero,such that,1.7 Linear Independence,Example,：,a.Determine if the set v,1,v,2,v,3,is linearly independent.,b.If possible,find a linear dependence relation among v,1,v,2,v,3.,Example,a.Row reduce the augmented matrix,Clearly,x,1,and x,2,are basic variables,and x,3,is free.,Each nonzero value of x,3,determines a nontrivial solution.,Hence,v,1,v,2,v,3,are linearly dependent.,Example 1,pletely row reduce the augmented matrix:,Thus,x,1,=2x,3,x,2,=-x,3,and x,3,is free.,Choose x,3,=5,Then x,1,=10 and x,2,=-5.,So one possible linear dependence relations among v,1,v,2,v,3,is,1.7 Linear Independence,The condition of linear independence:,For Matrix Columns,-,if and only if the equation,A,x,=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,=v,1,v,p,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,v,1,0,then some v,j,is a linear combination of the preceding vectors,v,1,v,j-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 v,1,v,p,in,R,n,is linearly dependent if,p,n,.,Theorem 9,If a set,S,=v,1,v,p,in,R,n,contains the zero vector,then 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 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,(c,u,)=c,T,(,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,(c,u,+d,v,)=c,T,(,u,)+d,T,(,v,),1.9 The Matrix of A Linear Transformation,Theorem 10,Let,T,:,R,n,R,m,be a linear transformation.Then there exists a unique matrix A such that,T,(,x,)=A,x,for all,x,in,R,n,In fact,A is the m,n matrix whose,j,th column is the vector T(e,j,),where e,j,is the,j,th column of the identity matrix in,R,n,:,A=T(e,1,)T(e,n,),1.9 The Matrix of A Linear Transformation,Example:,Find the standard matrix A for the dilation transformation,T,(,x,)=3,x,for,x,in R,2,Solution:,1.9 The Matrix of A Linear Transformation,2.Geometric Linear Transformations of R,2,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,Definition,(1),A mapping,T,:R,n,-R,m,is said to be,onto,R,m,if each,b,in R,m,is the image of,at least one,x,in R,n,(,满射,),(2),A mapping,T,:R,n,-R,m,is said to be,one-to-one,if each,b,in R,m,is the image of,at most one,x,in R,n,(,单射,),1.9 The Matrix of A Linear Transformation,1.9 The Matrix of A Linear Transformation,Theorem 11,Let,T,:R,n,-R,m,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,:R,n,-R,m,be a linear transformation and let,A,be the standard matrix for,T,.Then:,a.,T,maps R,n,onto R,m,if and only if the columns of,A,span R,m,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 B,Scalar Multiples,c,A: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,m,n,matrix,and if B is an,n,p,matrix with columns,b,1,b,p,then the product,AB,is the,m,p,matrix whose columns are A,b,1,A,b,p.,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,m,n,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,m,n,matrix and if,k,is a positive integer,then A,k,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=I,n,),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 R,n,the equation Ax=b has the unique solution x=A,-1,b.,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,-1,is 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 A,T,and the inverse of A,T,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 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 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 R,n,h.The columns of A span R,n,.,i.The linear transformation xAx maps R,n,onto R,n,.,j.There is an nn matrix C such that CA=I.,k.There is an nn matrix D such that AD=I.,l.A,T,is an invertible matrix.,2.4 Partitioned Matrices,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 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.,c,A: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 n,p,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 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