线性代数 chapter1 Linear Equations in Linear Algebra

1、 Linear Algebra and Its ApplicationWhat is Linear Algebra?pdevelops from the idea of trying to solve and analyze systems of linear equations.ptheory of matrices and determinants arise from this effort.pintricately linked with computer science .Why is Linear Algebra interesting?pIt has many applicati

2、ons in many diverse fields.nComputer graphicsnChemistrynEconomicsnBusinessnpIt strikes a nice balance between computation and theory.pGreat area in which to use technology (MatLab).How to Study Linear Algebra?pStudy before you start to work on exercises.p Prepare for each class period as you would f

3、or a language class.p Review frequently.GradingpClass Participation 10%pWeekly Written Assignments 30%pFinal Examination 60%Contents1、Linear Equations in Linear Algebra2、 Matrix Algebra3、 Determinants（行列式）（行列式）4 、Vector Spaces 5 、Eigenvalues and Eigenvectors（特征向量）（特征向量）6 、Orthogonality（正交性）（正交性） and

4、 Least Squares7 、Symmetric Matrices and Quadratic FormsCHAPTER 1Linear Equations in Linear AlgebraChapter 1 Linear Equation in Linear Algebra1.1 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 1.6

5、Application of Linear Systems 1.7 Linear Independence 1.8 Introduction to Linear Transformation 1.9 The Matrix of a Linear Transformation 1.10 Linear Modles 1.1 Systems of Linear EquationspWhat is a linear equation?pMatrix NotationpSolving a Linear SystempExistence and Uniqueness Questions1.1 System

6、s of Linear EquationspWhat is a linear equation? A linear equation in the variables x1,xn is an equation of the form a1x1 + a2x2+ . . . + anxn = b (1) where b and the coefficients a1,an are real or complex numbers. eg.1.1 Systems of Linear EquationspWhat is a system of linear equations? A system of

7、linear equations (or linear system) is a collection of one or more linear equations involving the same variables- x1, xna1,1x1+ a1,2x2+ . . . + a1,nxn = b1a2,1x1+ a2,2x2+ . . . + a2,nxn = b2. . .am,1x1+ am,2x2+ . . . + am,nxn = bm1.1 Systems of Linear EquationspA solution of the system of linear equ

8、ations - An assignment of values to the variables that satisfies (is a solution to) all of the equations in the system.nThe set of all possible solution is called the solution set.nTwo linear systems are equivalent if they have the same solution set.1.1 Systems of Linear EquationspA system of linear

9、 equations has either 1. No solution, or2. Exactly one solution, or3. Infinitely many solutions.consistentinconsistent1.1 Systems of Linear EquationsFig(a). Exactly one solutionFig(b). no solution Fig(c). Infinitely many solutions 1.1 Systems of Linear EquationspWhat is a linear equation?pMatrix Not

10、ationpSolving a Linear SystempExistence and Uniqueness Questions1.1 Systems of Linear EquationspMatrix NotationCoefficient matrixaugmented matrixThe size of a Matrix: how many rows and columns it has. 1.1 Systems of Linear EquationspWhat is a linear equation?pMatrix NotationpSolving a Linear Systemp

11、Existence and Uniqueness Questions1.1 Systems of Linear EquationspSolving a Linear SystemnTo replace one system with an equivalent system (one with the same solution set) that is easier to solvenThree basic operations to simplify a linear system 1. replace one equation by the sum of itself and a mul

12、tiple of another equation 2. interchange two equations 3. multiply all the terms in an equation by a nonzero constantp Solving a Linear System ：4*eq.1+eq.3()*eq.23*eq.2+eq.34*eq.3+eq.2-1*eq.3+eq.1Sol:。Upper triangular Back subsitutionWe strongly advise you always to check solutions! Did you get trie

13、d of writing the x1,x2,x3,and the = in the solution just illustrated? Only the coefficients change from one stage to another, so it is only the coefficients we really have to write down. Lets solve the above system again, by different means.p Solving a Linear System ：Sol:augmented matrix1.1 Systems

14、of Linear Equations 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. 1.1 Systems of Linear EquationspWhat is a linear equation?pMatri

15、x NotationpSolving a Linear SystempExistence and Uniqueness Questions1.1 Systems of Linear EquationspExistence and Uniqueness QuestionsnTwo 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;

16、 that is, is the solution unique?pEg: Determine if the following system is consistentSol: From example1, we have We know x3, and substitute the value of x3 into eq.2 could get x2 , then could determine x1 from eq.1. So a solution exists; the system is consistent.pEg:Determine if the following system

17、 is consistent:Sol: The equation 0 x1+0 x2+0 x3=(5/2) is never true, so the system is inconsistent.1.2 Row Reduction and Echelon FormspDefinitionpPivot PositionspThe Row Reduction AlgorithmpSolution of Linear SystemspParametric Descriptions of Solution SetspBack-SubstitutionpExistence and Uniqueness

18、 Questions1.2 Row Reduction and Echelon FormspDefinition： A rectangular matrix is in echelon form (or row echelon form) if : 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a c

19、olumn below a leading entry are zeros.*leading entry: the first nonzero entry in a nonzero row. pThe following matrices are in echelon form(upper triangular matrix):1.2 Row Reduction and Echelon FormspDefinition： A rectangular matrix is in reduced echelon form (or row reduced echelon form) if : 1. A

20、ll nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros. 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry

21、 in its column.pThe following matrices are in reduced echelon form:1.2 Row Reduction and Echelon FormspTheorem 1 : Uniqueness of the Reduced Echelon Form If a matrix A is row equivalent to an echelon matrix U, we call U an echelon form of A; If U is in reduced echelon form, we call U the reduced ech

22、elon form of A.1.2 Row Reduction and Echelon FormspDefinitionpPivot PositionspThe Row Reduction AlgorithmpSolution of Linear SystemspParametric Descriptions of Solution SetspBack-SubstitutionpExistence and Uniqueness Questions1.2 Row Reduction and Echelon FormspImportant Termsl pivot: A pivot in a r

23、ow echelon matrix U is a leading nonzero entry in a nonzero row.l pivot position: a position of a leading entry in an echelon form of the matrix.l pivot column: a column that contains a pivot position. Sol:Interchange row1 and row4Adding multiples of the first rows below:Example: Row reduce the matr

24、ix A below to echelon form, and locate the pivot columns of A. Adding -5/2 times row 2 to row3, and add 3/2 times row 2 to row 4 interchange rows 3 and 4 Note： There is no more than one in any row. There is no more than one in any colomn.1.2 Row Reduction and Echelon FormspThe Row Reduction Algorith

25、m 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

26、rows to modify. Step5 Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot.pExample: Transform the following matrix into reduced echelon:Sol:Step1:Step2:Step3:Step4:Step5:(1)(2)(3)(4) The combination of steps 1-4 is called the forward phase of the row

27、reductions algorithm. Steps 5 is called backward phase.(1)(2)1.2 Row Reduction and Echelon FormspSolution of Linear SystemsnThe row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system

28、.n Basic variable: any variable that corresponds to a pivot column in the augmented matrix of a system.n free variable：all nonbasic variables.pExample: Find the general solution of the following linear systemSol: The associated system now is The general solution is:1.2 Row Reduction and Echelon Form

29、spParametric Descriptions of Solution SetsnSolving a system amounts to finding a parametric description of the solution set or determine that the solution set is empty.nThe solution has many parametric descriptions.nWe make the arbitrary convention of always using the free variables as the parameter

30、s for describing a solution set.1.2 Row Reduction and Echelon FormspBack-SubstitutionnA computer program would solve system by back-substitution nRecommend use only the reduced echelon form to solve a system1.2 Row Reduction and Echelon FormspExistence and Uniqueness Questions Theorem 2 Existence an

31、d Uniqueness TheorempExample: Determine the existence and uniqueness of the solution to the systemSol. The basic variables are x1,x2,and x5; the free variables are x3 and x4. There is no equation such as 0 = 1, so the existence of a solution is already clear. Also the solution is not unique because

32、there are free variables.Using Row Reduction to Solve A Linear System1: Write the augmented matrix of the system.2: Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. If the system is inconsistent, Stop.3: Continue row reduction to obtain the reduced echelon fo

33、rm.4: Write the system of equations corresponding to the matrix obtained in step3.5: Rewrite each nonzero equation form step4 so that its one basic variable is expressed in terms of any free variables appearing in the equation.1.3 Vector Equationsp Vectors in R2p Geometric Description of R2p Vectors

34、 in R3p Vectors in R3p Linear Combinationp A Geometric Description of Spanv and Spanu,vp Linear Combinations in Applications1.3 Vector EquationsA matrix with only one column is called a column vector, or simply a vector. p Vectors in R2A two-dimensional vector is a pair of numbers, surrounded by bra

35、ckets. For example, 1.3 Vector Equationsp Vectors in R2u Notation: Different people use different notation for vector. v (boldface), (use arrows)u vectors are equal: If and only if they have the same corresponding entries. eg: = vuGeometric Description of R2 Vector as points Vectors with arrowsu Vec

36、tor Addition: We add vectors in the obvious way, componentwise:u Scalar Multiplication:Notes: the vector cv has the same direction as v if c 0 and the direction opposite to v if c n. pTheorem 9 If a set S = v1,vp in Rn contains the zero vector, then the set is linearly dependent.pDetermine by inspec

37、tion if the given set is linearly dependentSol.a. 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 Independent1.8 Linear TransformationspTransformationspMatrix TransformationspLinear Transformationsp1. Transforma

38、tionsAx = bAu = 0Fig. Transforming vectors1.8 Linear TransformationspTransformation T - Rn domain of T(定义域定义域) - Rm codomain of T(余定义域余定义域) - T: Rn Rm - Image of x T(x) in Rm（像）（像） - Range of T Set of all images T(x) range of T(值域值域)1.8 Linear Transformationsp Matrix Transformation a. Find T(u), the

39、 image of u under the transformation T. b. Find an x in R2 whose image under T is b. c. Is there more than one x whose image under T is b? d. Determine if c is in the range of the transformation T. LetExample：pSol.a. Compute b. Solve T(x) =b for x. (1) Hence, (2) c. Any x whose image under T is b mu

40、st satisfy (1). From (2), it is clear that equation (1) has a unique solution. So there is exactly one x whose image is b.d. The system is inconsistent. So c is not in the range T.1.8 Linear TransformationsE.g. Projection Transformation（投影变换）（投影变换）1.8 Linear TransformationsProjection Transformation1

41、.8 Linear Transformations- Shear transformation（错切变换）（错切变换）E.g. The image of the point is is1.8 Linear Transformations1.8 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

42、 all scalars c. Every Matrix transformation is a linear transformation.1.8 Linear TransformationspExample : Given a scalar r, define T: R2-R2 by T(x)=r x. T is called a contraction when 0 r 1 and a dilation when r 1. Let r = 3, and show that T is a linear Transformation.Sol. Let u,v be in R2 and let

43、 c,d be scalars. Then Thus T is a linear transformation.Fig. A dilation transformation（拉伸变换）1.8 Linear Transformations1.8 Linear TransformationspExample : Let T: R2 R2 be a linear transformation that maps into and maps into use the fact that T is linear to Find the images under T of 3U, 2v and 3U+2V

44、. Sol. 1.8 Linear TransformationspExample : Define a linear transformation T: R2-R2 by Find the image under T of Sol.ExampleFig. A rotation transformationExample ： A Company manufactures two products B and C. For $1.00 worth of product B, the company spend $.45 on materials, $.25 on labor, and $.15

45、on overhead. For $1.00 worth of product C, the company spend $.40 on materials, $.35 on labor, and $.15 on overhead. We construct a “unit cost ” matrix, U=b c, whose columns describe the “cost per dollar of output” for the products: Let x=(x1, x2) be a “production” vector, corresponding to x1 dollar

46、s of product B and x2 dollars of product C, and define T: R2R3 by The mapping T transforms a list of production quantities (measured In dollars ) into a list of total costs. The linearity of this mapping is reflected in two ways: (1) If production is increased by a factor of, say, 4, from x to 4x, t

47、hen the cost will increase by the same factor, from T(x) to 4T(x).(2) If x and y are production vectors, then the total cost vector associated with the combined production x+y is precisely the sum of the cost vectors T(X) and T(y).1.9 The Matrix of A Linear TransformationpThe Matrix of A Linear Tran

48、sformationpGeometric Linear Transformation of R2pExistence and Uniqueness QuestionspThe Matrix of A Linear Transformationnevery linear transformation from Rn to Rm is actually a matrix transformation x Ax and that important properties of T are intimately related to familiar properties of A. nThe key

49、 to finding A is to observe that T is completely determined by what it does to the columns of the nn identity matrix In.1.9 The Matrix of A Linear TransformationExample ：The columns of areSuppose T: R2-R3 Find a formula for the image of an arbitrary x in R2 . Sol:1.9 The Matrix of A Linear Transform

50、ation 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 jth column is the vector T(ej), where ej is the jth column of the identity matrix in Rn:A = T(e1) T(en) The matrix A is called the stand

51、ard matrix for the linear transformation T.1.9 The Matrix of A Linear TransformationpExample : Find the standard matrix A for the dilation transformation T(x) = 3x, for x in R2Sol.pExample : let T: R2R2 be the transformation that rotates each point in R2 about the origin through an angle , with coun

52、ter clockwise rotation for a positive angle. We could show geometrically that such a transformation is linear. Find the standard matrix A of this transformationSol. rotates into ，rotates intoBy Theorem 101.9 The Matrix of A Linear TransformationpGeometric Linear Transformations of R21.9 The Matrix of A Linear TransformationExpansions and Compressions（收缩变换和拉伸变换）（收缩变换和拉伸变换）1.9 The