linear algebra and its application习题答案线性代数chapter1

1、 Linear Algebra and Its Application ChengYing Gao Sun Yat-Sen University Automn 2009 What 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 . Wh

2、y is Linear Algebra interesting? pIt has many applications in many diverse fields. nComputer graphics nChemistry nEconomics nBusiness n pIt 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

3、work on exercises. p Prepare for each class period as you would for a language class. p Review frequently. Grading pClass Participation 10% pWeekly Written Assignments 30% pFinal Examination 60% Contents 1、Linear Equations in Linear Algebra 2、 Matrix Algebra 3、 Determinants（行列式）（行列式） 4 、Vector Space

4、s 5 、Eigenvalues and Eigenvectors（特征向量）（特征向量） 6 、Orthogonality（正交性）（正交性） and Least Squares 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

5、 Equation 1.4 The Matrix Equation Ax = b 1.5 Solution Sets of Linear Systems 1.6 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 Equations pWhat is a linear equation? pM

6、atrix Notation pSolving a Linear System pExistence and Uniqueness Questions 1.1 Systems of Linear Equations pWhat 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

7、. eg. 1.1 Systems of Linear Equations pWhat is a system of linear equations? A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables- x1, xn a1,1x1+ a1,2x2+ . . . + a1,nxn = b1 a2,1x1+ a2,2x2+ . . . + a2,nxn = b2 . . . am,1x1+ am,2

8、x2+ . . . + am,nxn = bm 1.1 Systems of Linear Equations pA solution of the system of linear equations - 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

9、 equivalent if they have the same solution set. 1.1 Systems of Linear Equations pA 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 Fig(a). Exactly one solution Fig(b). no soluti

10、on Fig(c). Infinitely many solutions 1.1 Systems of Linear Equations pWhat is a linear equation? pMatrix Notation pSolving a Linear System pExistence and Uniqueness Questions 1.1 Systems of Linear Equations pMatrix Notation Coefficient matrixaugmented matrix The size of a Matrix: how many rows and c

11、olumns it has. 1.1 Systems of Linear Equations pWhat is a linear equation? pMatrix Notation pSolving a Linear System pExistence and Uniqueness Questions 1.1 Systems of Linear Equations pSolving a Linear System nTo replace one system with an equivalent system (one with the same solution set) that is

12、easier to solve nThree basic operations to simplify a linear system 1. replace one equation by the sum of itself and a multiple of another equation 2. interchange two equations 3. multiply all the terms in an equation by a nonzero constant p Solving a Linear System ： 4*eq.1+eq.3 ()*eq.2 3*eq.2+eq.3

13、4*eq.3+eq.2 -1*eq.3+eq.1 Sol: 。 Upper triangular Back subsitution We strongly advise you always to check solutions! Did you get tried 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 h

14、ave to write down. Lets solve the above system again, by different means. p Solving a Linear System ： Sol: augmented matrix 1.1 Systems 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

15、rows. 3. (Scaling) Multiply all entries in a row by a nonzero constant. 1.1 Systems of Linear Equations pWhat is a linear equation? pMatrix Notation pSolving a Linear System pExistence and Uniqueness Questions 1.1 Systems of Linear Equations pExistence and Uniqueness Questions nTwo fundamental quest

16、ions 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? pEg: Determine if the following system is consistent Sol: From example1, we have We know x3, and substitute the value of x3

17、 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 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 Forms pDefinition pPivo

18、t Positions pThe Row Reduction Algorithm pSolution of Linear Systems pParametric Descriptions of Solution Sets pBack-Substitution pExistence and Uniqueness Questions 1.2 Row Reduction and Echelon Forms pDefinition： A rectangular matrix is in echelon form (or row echelon form) if : 1. All nonzero row

19、s 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. *leading entry: the first nonzero entry in a nonzero row. pThe following matrices are in echelon form(up

20、per triangular matrix): 1.2 Row Reduction and Echelon Forms pDefinition： A rectangular matrix is in reduced echelon form (or row reduced 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

21、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 in its column. pThe following matrices are in reduced echelon form: 1.2 Row Reduction and Echelon Forms pTheorem 1 : Uniqueness of the Redu

22、ced 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 echelon form of A. 1.2 Row Reduction and Echelon Forms pDefinition pPivot Positions pThe Row Reduction Algorithm pSolution of Linear Systems

23、 pParametric Descriptions of Solution Sets pBack-Substitution pExistence and Uniqueness Questions 1.2 Row Reduction and Echelon Forms pImportant Terms l pivot: A pivot in a row echelon matrix U is a leading nonzero entry in a nonzero row. l pivot position: a position of a leading entry in an echelon

24、 form of the matrix. l pivot column: a column that contains a pivot position. Sol: Interchange row1 and row4 Adding multiples of the first rows below: Example: Row reduce the matrix 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 t

25、o 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 Forms pThe Row Reduction Algorithm Step1 Begin with the leftmost nonzero column. Step2 Select a nonzero entry in the pivot column as a pivot. Step3 Use

26、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 pi

27、vot. 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 reductions algorithm. Steps 5 is called backward phase. (1) (2) 1.2 Row Reduction and Echelon Forms pSolutio

28、n of Linear Systems nThe 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. n Basic variable: any variable that corresponds to a pivot column in the augmented matrix of a system

29、. n free variable：all nonbasic variables. pExample: Find the general solution of the following linear system Sol: The associated system now is The general solution is: 1.2 Row Reduction and Echelon Forms pParametric Descriptions of Solution Sets nSolving a system amounts to finding a parametric desc

30、ription 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 parameters for describing a solution set. 1.2 Row Reduction and Echelon Forms pBack-Substitution nA com

31、puter program would solve system by back-substitution nRecommend use only the reduced echelon form to solve a system 1.2 Row Reduction and Echelon Forms pExistence and Uniqueness Questions Theorem 2 Existence and Uniqueness Theorem pExample: Determine the existence and uniqueness of the solution to

32、the system Sol. 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 there are free variables. Using Row Reduction to Solve A Linear System 1: Write the aug

33、mented 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 form. 4: Write the system of equations corresponding to the matrix obtained in step3.

34、 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 Equations p Vectors in R2 p Geometric Description of R2 p Vectors in R3 p Vectors in R3 p Linear Combination p A Geometric Description of Span

35、v and Spanu,v p Linear Combinations in Applications 1.3 Vector Equations A matrix with only one column is called a column vector, or simply a vector. p Vectors in R2 A two-dimensional vector is a pair of numbers, surrounded by brackets. For example, 1.3 Vector Equations p Vectors in R2 u Notation: D

36、ifferent 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: = uGeometric Description of R2 Vector as points Vectors with arrows u Vector Addition: We add vectors in the obvious way, componentwise: u S

37、calar 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 inspection if the given set is linearly dependent Sol. a. The set cont

38、ains 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 pTransformations pMatrix Transformations pLinear Transformations p1. Transformations Ax = b Au = 0 Fig. Transforming vectors 1.8 Linea

39、r Transformations pTransformation 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 Transformations p Matrix Transformation a. Find T(u), the image of u under the transformation T. b. Find

40、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. Let Example： pSol. a. Compute b. Solve T(x) =b for x. (1) Hence, (2) c. Any x whose image under T is b must satisfy (1). From (2), it is clear that eq

41、uation (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 Transformations E.g. Projection Transformation（投影变换）（投影变换） 1.8 Linear Transformations Projection Transformation 1.8 Linear Transformations - Shear trans

42、formation（错切变换）（错切变换） E.g. The image of the point is is 1.8 Linear Transformations 1.8 Linear Transformations pLinear 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. Every Matrix trans

43、formation is a linear transformation. 1.8 Linear Transformations pExample : 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 c,d be scalars. Then Thus T is

44、 a linear transformation. Fig. A dilation transformation（拉伸变换） 1.8 Linear Transformations 1.8 Linear Transformations pExample : 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. Sol. 1.8 Linear Transform

45、ations pExample : Define a linear transformation T: R2-R2 by Find the image under T of Sol. Example Fig. A rotation transformation Example ： 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 on overhead. For $1.00

46、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 dollars of product B and x2 d

47、ollars 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, then the cost will incre

48、ase 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 Transformation pThe Matrix of A Linear Transformation pGeometri

49、c Linear Transformation of R2 pExistence and Uniqueness Questions pThe Matrix of A Linear Transformation nevery 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 to finding A is

50、 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 Transformation Example ：The columns of are Suppose T: R2-R3 Find a formula for the image of an arbitrary x in R2 . Sol: 1.9 The Matrix of A Linear Transformation pTheor

51、em 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 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 standard matrix

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

53、ockwise rotation for a positive angle. We could show geometrically that such a transformation is linear. Find the standard matrix A of this transformation Sol. rotates into ，rotates into By Theorem 10 1.9 The Matrix of A Linear Transformation pGeometric Linear Transformations of R2 1.9 The Matrix of A Linear Transformation Expansions and Compressions （收缩变换和拉伸