中山大学-线性代数期末总复习ppt课件.ppt
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1、 REVIEW FOR THE FINAL EXAMGao ChengYingSun Yat-Sen UniversitySpring 2007Linear Algebra and Its ApplicationREVIEW FOR THE FINAL EXAMChapter 1 Linear Equations in Linear AlgebraChapter 2 Matrix AlgebraChapter 3 Determinants Chapter 4 Vector SpacesChapter 5 Eigenvalues and EigenvectorsChapter 6 Orthogo
2、nality and Least SquaresChapter 7 Symmetric Matrices and Quadratic FormsCHAPTER 1Linear Equations in Linear AlgebraChapter 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 L
3、inear Systems 1.7 Linear Independence 1.8 Introduction to Linear Transformation 1.9 The Matrix of a Linear Transformation1.1 Systems of Linear Equationsp1. linear equation a1x1 + a2x2+ . . . + anxn = b Systems of Linear Equationsmnmnmmnnnnbxaxaxabxaxaxabxaxaxa221122222121112121111.1 Systems of Linea
4、r EquationspConfficient matrix and augmented matrixCoefficient matrixaugmented matrixmnmmnnaaaaaaaaa212222111211mmnmmnnbaaabaaabaaa212222211112111.1 Systems of Linear EquationspA solution to a system of equations nA system of linear equations has either 1. No solution, or2. Exactly one solution, or3
5、. Infinitely many solutions.consistentinconsistent1.1 Systems of Linear EquationspSolving a Linear SystemnElementary 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 b
6、y a nonzero constant.Examplesp 1. Solving a Linear Systemp 2. Discuss the solution of a linear system which has unknown variable1.1 Systems of Linear EquationspExistence and Uniqueness QuestionsnTwo fundamental questions about a linear system1. Is the system consistent; that is, does at least one so
7、lution exist?2. If a solution exists, is it the only one; that is, is the solution unique?1.2 Row Reduction and Echelon FormspThe following matrices are in echelon form:pThe following matrices are in reduced echelon form:pivot position1.2 Row Reduction and Echelon FormspTheorem 1 Uniqueness of the R
8、educed Echelon Form1.2 Row Reduction and Echelon FormspThe 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 t
9、o 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 FormspSolution of Linear Systems (Using Row Reduction) eg. Find th
10、e general solution of the following linear systemSolution:1.2 Row Reduction and Echelon Forms The associated system now is The general solution is:1.2 Row Reduction and Echelon FormspTheorem 2 Existence and Uniqueness Theorem1.3 Vector EquationspAlgebraic Properties of For all u, v, w in and all sca
11、lars c and d: where u denotes (-1)u1.3 Vector EquationspSubset of - Span v1,vp is collection of all vectors that can be written in the form with c1,cp scalars.1.4 The Matrix Equation Ax = bp1.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
12、by Ax, 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 = bpTheorem 3 If A is an mn matrix, with column a1,an, and if b is in Rm, the matrix equationAx = b has the same solution set as the vector equation which, in turn
13、, has the same solution set as the system of linear equation whose augmented matrix is 1.4 The Matrix Equation Ax = bp2. 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
14、? 1.4 The Matrix Equation Ax=bpSolution Row reduce the augmented matrix for Ax=b:p The equation Ax=b is not consistent for every b.= 0 (for some choices of b)1.4 The Matrix Equation Ax=bpTheorem 4 Let A be an mn matrix. Then the following statements are logically equivalent. That is, for a particula
15、r A, either they are all true statements or they are all false. a. For each b in Rm, the equation 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=bp3. Computation of Ax
16、Example . Compute Ax, whereSolution.1.4 The Matrix Equation Ax=bp4. Properties of the Matrix-Vector Product AxTheorem 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 Systemsp1. Solution of Homogeneous Linear Systems p2. Solution of Nonhomogeneous
17、 Systems1.5 Solution Set of Linear Systemsp1. Homogeneous Linear Systems Ax = 0 - trivial solution (平凡解) - nontrivial solution (非平凡解)pThe 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 SystemspExample Solve
18、the Homogeneous Linear Systems pSolution (1) Row reductionExamplep(2) Row reduction to reduced echelon formp(3) The general solution1.5 Solution Set of Linear Systemsp2. Solution of Nonhomogeneous Systems eg. Describe all solutions of Ax=b, whereSolution 1.5 Solution Set of Linear SystemsThe general
19、 solution of Ax=b has the formThe solution set of Ax=b in parametric vector form1.5 Solution Set of Linear SystemspTheorem 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 a
20、ny solution of the homogeneous equation Ax=0.1.7 Linear Independencep1. Definition - 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 i
21、f there exist weights c1,cp , not all zero, such that1.7 Linear IndependencepExample: a. Determine if the set v1,v2,v3 is linearly independent. b. If possible, find a linear dependence relation among v1,v2,v3.Example pa. Row reduce the augmented matrix Clearly, x1 and x2 are basic variables, and x3
22、is free. Each nonzero value of x3 determines a nontrivial solution. Hence v1,v2,v3 are linearly dependent.Example 1pb. 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.
23、7 Linear IndependenceThe condition of linear independence: pFor Matrix Columns - if and only if the equation Ax=0 has only the trivial solution.pFor Sets of One or Two Vectors - if and only if neither of the vectors is a multiple of the other.pFor Sets of Two or More Vectors - Theorem 7 (Characteriz
24、ation of Linearly Dependent Sets)p4. Linear Independence of Sets of Two or More Vectors 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.
25、 In fact, if S is linearly dependent and v10, then some vj is a linear combination of the preceding vectors, v1,vj-1.b1.7 Linear IndependencepTheorem 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 d
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