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1、Li Jie1 1Introduction toLinear AlgebraLee W.JohnsonR.Dean RiessJimmy T.Arnold 123456Li Jie2 2OrganizationChapter oneMatrices and systems of linear equationsChapter twoVectors in 2-space and 3-space(optional)Chapter threeThe vector space RnChapter fourThe eigenvalue problemChapter fiveVector spaces a
2、nd linear transformationsChapter sixDeterminantsChapter sevenEigenvalues and applicationsLi Jie3 31 Matrices and systems of linear equationsLi Jie4 4Overview We next introduce the operations of addition and multiplication for matrices and show how these operations enable us to express a linear syste
3、m in matrix-vector terms as AX=b.In this chapter we discuss systems of linear equations and methods(such as Gauss-Jordan elimination)for solving these systems.We introduce matrices as a convenient language for describing systems and the Gauss-Jordan solution method.Li Jie5 5Core sectionsIntroduction
4、 to matrices and systems of linear equationsEchelon form and Gauss-Jordan eliminationConsistent systems of linear equationsMatrix operations Algebraic properties of matrix operationsLinear independence and nonsingular matricesMatrix inverses and their propertiesLi Jie6 61.1 Introduction To Matrices
5、And Systems Of Linear EquationsA linear equation in n unknowns is an equation that can be put in the formThe coefficients a1,a2,an and the constant b are known,and x1,x2,xn denote the unknowns.A equation is called linear because each term has degree one in the variables x1,x2,xn.Otherwise the equati
6、on is called nonlinear.Li Jie7 7Example1:Which of the following equations are linear?Li Jie8 8An(m n)system of linear equations is a set of equations of the form:A solution to system(*)is a sequence s1,s2,sn of numbers that is simultaneously a solution(联立解）for each equation in the system.The double
7、subscript notation used for the coefficients is necessary to provide an“address”for each coefficient.For example,a32 appears in the third equation as the coefficient of x2.Li Jie9 91.Geometric interpretations of solution sets(1)(2 2)system of linear equations.1.The two lines are coincident(the same
8、line),so there are infinitely many solutions.2.The two lines are parallel(never meet),so there are no solutions.3.The two lines intersect at a single point,so there is a unique solution.Li Jie1010(2)(2 3)system of linear equations.1.The two planes might be coincident.In this case,the system has infi
9、nitely many solutions.2.The two planes might be parallel.In this case,the system has no solution.3.The two planes might intersect in a line.In this case,the system has infinitely many solutions.Li Jie1111(3)(3 3)system of linear equations.1.The three planes might be coincident,or intersect in a line
10、.Then the system has infinitely many solutions.2.The three planes are parallel,there are two planes be parallel,or the three planes intersect three lines which for every two lines are parallel.Then the system has no solution.3.The three planes intersect at a single point.In this case,the system has
11、a unique solution.Li Jie1212Remark:An(m n)system of linear equations has either infinitely many solutions,no solution,or a unique solution.In general,a system of equations is called consistent if it has at least one solution,and the system is called inconsistent if it has no solution.Li Jie13132.Mat
12、rices We begin our introduction to matrix theory by relating matrices to the problem of solving systems of linear equations.Initially we show that matrix theory provides a convenient and natural symbolic language to describe linear systems.Later we show that matrix theory is also an appropriate and
13、powerful framework within which to analyze and solve more general linear problems,such as least-squares approximations,representations of linear operations,and eigenvalue problems.Li Jie1414More generally,an(m n)matrix is a rectangular array of numbers of the formThus an(m n)matrix has m rows and n
14、columns.The subscripts for the entry aij indicate that the number appears in the ith row and jth column of A.Li Jie15153.Matrix representation of a linear systemThe coefficient matrix for the system is a(m n)matrix A:Li Jie1616The augmented matrix（增广矩阵）for the system is a m(n+1)matrix B which is usu
15、ally denoted as A|b,where A is the coefficient matrix and b=b1 b2 bmT.Li Jie17174.Elementary operations(初等变换）As we shall see,there are two steps involved in solving an(m n)system of equations.1.Reduction of the system(that is,the elimination of variables).2.Description of set of solutions.Definition
16、 1.1.1:two systems of linear equations in n unknowns are equivalent provided that they have the same set of solutions.等价等价Li Jie1818Elementary Operations:1.Interchange two equations.2.Multiply an equation by a nonzero scalar.3.Add a constant multiple of one equation to another.NotationElementary ope
17、ration performedEiEjThe ith and jth equations are interchanged.kEiThe ith equation is multiplied by the nonzero scalar k.Ei+kEjk times the jth equation is added to the ith equation.Li Jie1919Theorem 1.1.1:If one of the elementary operations is applied to a system of linear equations then the resulti
18、ng system is equivalent to the original system.Example2:Use elementary operations to solve the systemLi Jie2020Solution:The elementary operation E2+E1 produces the following equivalent system:The operation 1/3 E2 then leads toFinally,using the operation E1-E2,we obtainThis method is called Gauss-Jor
19、dan elimination.Li Jie21215.Row Operations:Definition1.1.2:The following operations,performed on the rows of a matrix,are called elementary row operations:1.Interchange two rows.2.Multiply a row by a nonzero scalar.3.Add a constant multiple of one row to another.Li Jie2222Notation Elementary Row Ope
20、ration RiRjThe ith and jth rows are interchanged.kRiThe ith row is multiplied by the nonzero scalar k.Ri+kRjk times the jth row is added to the ith row.We say that two(m n)matrices,B and C,are row equivalent if one can be obtained from the other by a sequence of elementary row operations.Li Jie23231
21、.Form the augmented matrix B for the system.2.Use elementary row operations to transform B to a row equivalent matrix C which represents a“simpler”system.3.Solve the simpler system that is represented by C.Now if B is the augmented matrix for a system of linear equations and if C is row equivalent t
22、o B,then C is the augmented matrix for an equivalent system.Thus,we can solve a linear system with the following steps:Li Jie2424Example3:Solution:Li Jie2525Li Jie2626Li Jie2727Corollary:Suppose A|b and C|d are augmented matrices,each representing a different(m n)system of linear equations.If A|b an
23、d C|d are row equivalent matrices,then the two systems are also equivalent.Li Jie28281.2 Echelon form and Gauss-Jordan eliminationGiven systemof equationsAugmentedmatrixReducedmatrixReduced systemof equationSolutionProcedure for solving a system of linear equationsLi Jie29291.Echelon Form（阶梯形（阶梯形 a
24、staircase-like pattern）Definition 1.2.1:An(m n)matrix B is in echelon form if:1.All rows that consist entirely of zeros are grouped together at the bottom of the matrix.2.In every nonzero row,the first nonzero entry(counting from left to right)is a 1.3.If the(i+1)-st row contains nonzero entries,the
25、n the first nonzero entry is in a column to the right of the first nonzero entry in the ith row.Li Jie3030Definition 1.2.2:A matrix that is in echelon form is in reduced echelon form provided that the first nonzero entry in any row is the only nonzero entry in its column.Example 1:For each matrix sh
26、own,choose one of the following phrases to describe the matrix.1.The matrix is not in echelon form.2.The matrix is in echelon form,but not in reduced echelon form.3.The matrix is in reduced echelon form.Li Jie3131Li Jie32322.Solving a linear system whose augmented matrix is in reduced echelon formEx
27、ample 2:Each of the following matrices is in reduced echelon form and is the augmented matrix for a system of linear equations.In each case,give the system of equations and describe the solution.Li Jie3333Solution:Matrix B is the augmented matrix for the following system:Therefore,the system has the
28、 unique solution x1=3,x2=-2,and x3=7.Matrix C is the augmented matrix for the following system:Because no values for x1,x2,or x3 can satisfy the third equation,the system is inconsistent.Li Jie34343.Recognizing an inconsistent systemTheorem1.2.1:Let A|b be the augmented matrix for an(m n)linear syst
29、em of equations,and let A|b be in reduced echelon form.If the last nonzero row of A|b has its leading 1 in the last column,then the system of equations has no solution.That is,the system represented by A|b is inconsistent.Li Jie3535Step 1.Create the augmented matrix for the system.Step 2.Transform t
30、he matrix in Step 1 to reduced echelon form.Step 3.Decode the reduced matrix found in Step 2 to obtain its associated system of equations.Step 4.By examining the reduced system in Step 3,describe the solution set for the original system.4.Solving a system of equationsLi Jie36365.Reduction to echelon
31、 formTheorem1.2.2:Let B be an(m n)matrix.There is a unique(m n)matrix C such that:(1)C is in reduced echelon form(2)C is row equivalent to B.Li Jie3737Reduction to reduced echelon form for an(m n)matrix:(1)Locate the first(left-most)column that contains a nonzero entry.(2)If necessary,interchange th
32、e first row with another row so that the first nonzero column has a nonzero entry in the first row.(3)If a denotes the leading nonzero entry in row one,multiply each entry in row one by 1/a.Li Jie3838(4)Add appropriate multiples of row one to each of the remaining rows so that every entry below the
33、leading 1 in row one is a 0.(5)Temporarily ignore the first row of this matrix and repeat(1)(4)on the submatrix that remains.Stop the process when the resulting matrix is in echelon form.(6)Having reached echelon form in(5),continue on to reduced echelon form as follows:Proceeding upward,add multipl
34、es of each nonzero row to the rows above in order to zero all entries above the leading 1.Li Jie3939Example3:Use elementary row operations to transform the following matrix to reduced echelon formExercise:Li Jie4040Example 4:Solve the following system of equations:Solution:transform the augmented ma
35、trix to reduced echelon form.R1+R2R2+R1R4-3R1Li Jie4141Then matrix above represents the following system of equationsR1+2R3R4+2R3R1-2R2R3+3R2R4-4R2Li Jie4242Remark:In Eq.(1)we have a nice description of all of the infinitely many solutions to the original systemit is called the general solution for
36、the system.For this example,x2 and x5 are independent variables and can be assigned values arbitrarily.The variables x1,x3,and x4 are dependent variables,and their values are determined by the values assigned to x2 and x5.Solving the preceding system,we find:Li Jie4343Exercises:P27 28,30,49,53Partic
37、ular solution.(omit)Three people play a game in which there are always two winners and one loser.They have the understanding that the loser gives each winner an amount equal to what the winner already has.After three games,each has lost just once and each has$24.With how much money did each begin?Li
38、 Jie44441.3 Consistent Systems of Linear Equations1.Solution possibilities for a consistent linear systemOur goal is to deduce as much information as possible about the solution set of system(1)without actually solving the system.Li Jie4545Theorem1.3.1:Let the matrix C|d is in reduced echelon form.T
39、he system represented by the matrix C|d is inconsistent if and only if C|d has a row of the form 0,0,0,1.Theorem1.3.2:Every variable corresponding to a leading 1 in C|d is a dependent variable.Theorem1.3.3:Let r denote the number of nonzero rows in C|d.Then,r n+1.Li Jie4646Theorem1.3.4:Let r denote
40、the number of nonzero rows in C|d.If the system represented by C|d is consistent,then r n.Theorem1.3.5:Let C|d be an m(n+1)matrix in reduced echelon form,where C|d represents a consistent system.Let C|d have r nonzero rows.Then r n and in the solution of the system there are n-r variables that can b
41、e assigned arbitrary values.Theorem 1.3.6:Consider an(m n)system of linear equations.If mn,then either the system is inconsistent or it has infinitely many solutions.Li Jie47472.Homogeneous Systems（齐次线性方程组)The(m n)system of linear equations given in(2)is called a homogeneous system of linear equatio
42、ns:A homogeneous system is always consistent,because x1=x2=xn=0 is a solution to system(2).This solution is called the trivial solution(平平凡凡解解）or zero solution,and any other solution is called a nontrivial solution.Li Jie4848Theorem1.3.7:A homogeneous(m n)system of linear equations always has infini
43、tely many nontrivial solutions when mm,then this set is linearly dependent.Definition1.7.2:An(n n)matrix A is nonsingular if the only solution to Ax=0 is x=0.Furthermore,A is said to be singular if A is not nonsingular.Theorem1.7.2:The(n n)matrix A=A1,A2,An is nonsingular if and only is A1,A2,An is
44、a linearly independent set.Theorem1.7.3:Let A be an(n n)matrix.The equation Ax=b has a unique solution for every(n 1)column vector b if and only if A is nonsingular.Exercises:P78 49,50Li Jie87871.9 Matrix Inverses And Their Properties1.The matrix inverseDefinition1.9.1:Let A be an(n n)matrix.We say
45、that A is invertible if we can find an(n n)matrix A-1 such thatThe matrix A-1 is called an inverse for A.Example1:Let find its inverse matrix.Li Jie88882.Using inverses to solve systems of linear equationsAX=b X=A-1bAX=B X=A-1Bwhere A is an(n n)matrix,B and X are(n m)matrices.Li Jie89893.Existence o
46、f inversesLemma:Let P,Q,and R be(n n)matrices such that PQ=R.If either P or Q is singular,then so is R.Theorem1.9.1:Let A be an(n n)matrix.Then A has an inverse if and only if A is nonsingular.Li Jie90904.Calculating the inverseComputation of A-1Step1.Form the(n 2n)matrix A|I.Step2.Use elementary ro
47、w operations to transform A|I to the form I|B.Step3.Reading form this final form,A-1=B.Li Jie9191Example1:solution：Li Jie9292 Example2:solution：Zero rownot inverseLi Jie9393ExampleLi Jie9494Example2:Find the inverse of the(n n)matrixExercise 1.Find the inverse of the(n n)matrix2.Find the the matrix
48、X such that AXB=C,whereLi Jie9595Theorem1.9.2:Let A be an(n n)matrix.Then A is nonsingular if and only if A is row equivalent to I.Li Jie96965.Properties of matrix inversesTheorem1.9.3:Let A and B be(n n)matrices,each of which has an inverse.Then:1.A-1 has an inverse,and(A-1)-1=A.2.AB has an inverse
49、,and(AB)-1=B-1A-1.3.If k is a nonzero scalar,then kA has an inverse,and(kA)-1=(1/k)A-1.4.AT has an inverse,and(AT)-1=(A-1)T.Li Jie9797Theorem1.9.4:Let A be an(n n)matrix.The following are equivalent:1.A is nonsingular;that is,the only solution of Ax=0.2.The column vectors of A are linearly independe
50、nt.3.Ax=b always has a unique solution.4.A has an inverse.5.A is row equivalent to I.Li Jie98986.Ill-conditioned matrixIn applications the equation Ax=b often serves as a mathematical model for a physical problem.In these cases it is important to know whether solutions to Ax=b are sensitive to small