《introductiontolinearalgebra教学课件》chapter1

1、Introduction to Linear Algebra Lee W.Johnson R.Dean Riess Jimmy T.Arnold Organization Chapter one Matrices and systems of linear equations Chapter two Vectors in 2-space and 3-space Chapter three The vector space Rn Chapter four The eigenvalue problem Chapter five Vector spaces and linear transforma

2、tions (optional) Chapter six determinants Chapter seven eigenvalues and applications (optional) Chapter 1 Matrices and Systems of Linear Equations Overview In this chapter we discuss systems of linear equations and methods (such as Gauss-Jordan elimination) for solving these systems. We introduce ma

3、trices as a convenient language for describing systems and the Gauss-Jordan solution method. We next introduce the operations of addition and multiplication for matrices and show how these operations enable us to express a linear system in matrix-vector terms as Ax=b. Core sections w Introduction to

4、 matrices and systems of linear equations w Echelon form and Gauss-Jordan elimination w Consistent systems of linear equations w Matrix operations w Algebraic properties of matrix operations w Linear independence and nonsingular matrices w Matrix inverses and their properties 1.11.1 Introduction to

5、matrices and Introduction to matrices and systems of linear equations systems of linear equations A linear equation in n unknowns is an equation that can be put in the form ) 1 (. 2211 bxaxaxa nn The coefficients a1 ,a2, an and the constant b are known, and x1,x2, xn denote the unknowns. A equation

6、is called linear because each term has degree one in the variables x1,x2, xn . Otherwise the equation is called nonlinear. Example1: Which of the following equations are linear? 37. 6 0. 5 cossin. 4 cossin. 3 1. 2 32. 1 21 21 2 2 1 2 21 1 2 1 2 21 221 31 xx xx xxxx xxxx xxx xx An (m n) system of lin

7、ear 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 for each equation in the system. The double subscript notation used for the coefficients is necessary to provide an “address” for each coefficient. For examp

8、le, a32 appears in the third equation as the coefficient of x2. mnmnmm nn nn bxaxaxa bxaxaxa bxaxaxa 2211 22222121 11212111 (*) 1. Geometric interpretations of solution sets 2222121 1212111 bxaxa bxaxa (1) (2 2) system of linear equations. The two lines are coincident (the same line), so there are i

9、nfinitely many solutions. The two lines are parallel (never meet), so there are no solutions. The two lines intersect at a single point, so there is a unique solution. 2323222121 1313212111 bxaxaxa bxaxaxa (2) (2 3) system of linear equations. The two planes might be coincident. In this case, the sy

10、stem has infinitely many solutions. The two planes might be parallel. In this case, the system has no solution. The two planes might intersect in a line. In this case, the system has infinitely many solutions. 3333232131 2323222121 1313212111 bxaxaxa bxaxaxa bxaxaxa (3) (3 3) system of linear equati

11、ons. The three planes might be coincident, or intersect in a line. Then the system has infinitely many solutions. 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. The th

12、ree planes intersect at a single point. In this case, the system has a unique solution. Remark: 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

13、 system is called inconsistent if it has no solution. 2. Matrices 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

14、. Later we show that matrix theory is also an appropriate and powerful framework within which to analyze and solve more general linear problems, such as least-squares approximations, representations of linear operations, and eigenvalue problems. More generally, an (m n) matrix is a rectangular array

15、 of numbers of the form mnmm n n aaa aaa aaa A 21 22221 11211 Thus an (m n) matrix has m rows and n columns. The subscripts for the entry aij indicate that the number appears in the ith row and jth column of A. 3. Matrix representation of a linear system mnmnmm nn nn bxaxaxa bxaxaxa bxaxaxa 2211 222

16、22121 11212111 The coefficient matrix for the system is a (m n) matrix A: mnmm n n aaa aaa aaa A 21 22221 11211 The augmented matrix for the system is a m (n+1) matrix B which is usually denoted as A|b, where A is the coefficient matrix and b=b1 b2 bmT. mmnmm n n baaa baaa baaa A 21 222221 111211 4.

17、 Elementary operations As we shall see, there are two steps involved in solving an (m n) system of equations. Reduction of the system (that is, the elimination of variables). 1. Description of set of solutions. Definition 1.1.1: two systems of linear equations in n unknowns are equivalent provided t

18、hat they have the same set of solutions. Elementary Operations: Interchange two equations. Multiply an equation by a nonzero scalar. 1. Add a constant multiple of one equation to another. Theorem 1.1.1: If one of the elementary operations is applied to a system of linear equations then the resulting

19、 system is equivalent to the original system. NotationElementary operation performed EiEjThe ith and jth equations are interchanged. kEi The ith equation is multiplied by the nonzero scalar k. Ei +kEj k times the jth equation is added to the ith equation. Example2: Use elementary operations to solve

20、 the system .42 5 21 21 xx xx Solution: The elementary operation E2+E1 produces the following equivalent system: .93 5 2 21 x xx The operation 1/3 E2 then leads to .3 5 2 21 x xx Finally, using the operation E1- E2, we obtain .3 2 2 1 x x This method is called Gauss-Jordan elimination. 5.Row Operati

21、ons: NotationElementary Row Operation RiRjThe ith and jth rows are interchanged. kRi The ith row is multiplied by the nonzero scalar k. Ri +kRjk times the jth row is added to the ith row. Definition1.1.2: The following operations, performed on the rows of a matrix, are called elementary row operatio

22、ns: 1.Interchange two rows. 2.Multiply a row by a nonzero scalar. 3.Add a constant multiple of one row to another. 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. Now if B is the augmented matrix for a syst

23、em of linear equations and if C is row equivalent to B, then C is the augmented matrix for an equivalent system. Thus, we can solve a linear system with the following steps: 1. Form the augmented matrix B for the system. 2. Use elementary row operations to transform B to a row equivalent matrix C wh

24、ich represents a “simpler” system. 3. Solve the simpler system that is represented by C. Example3: 2242 1553 22 321 321 32 xxx xxx xx Solution: 2242 1553 22 :System 321 321 32 xxx xxx xx 2242 1553 2120 :MatrixAugmented 22 1553 2242 : 32 321 321 31 xx xxx xxx EE 2120 1553 2242 : 31 RR 22 1553 12 :)2/

25、1( 32 321 321 1 xx xxx xxx E 2120 1553 1121 :)2/1( 1 R 22 22 12 :3 32 32 321 12 xx xx xxx EE 2120 2210 1121 :3 12 RR 22 22 12 :)1( 32 32 321 2 xx xx xxx E 2120 2210 1121 :)1( 2 R 22 22 35 :2 32 32 31 21 xx xx xx EE 2120 2210 3501 :2 21 RR 63 22 35 :2 3 32 31 23 x xx xx EE 6300 2210 3501 :2 23 RR 2 2

26、2 35 :)3/1( 3 32 31 3 x xx xx E 2100 2210 3501 :)3/1( 3 R 2 22 7 :5 3 32 1 31 x xx x EE 2100 2210 7001 :5 31 RR 2 2 7 :2 3 2 1 32 x x x EE 2100 2010 7001 :2 32 RR Corollary: Suppose A|b and C|d are augmented matrices, each representing a different (m n) system of linear equations. If A|b and C|d are

27、 row equivalent matrices, then the two systems are also equivalent. 1.21.2 Echelon form and Echelon form and Gauss-Jordan eliminationGauss-Jordan elimination Given system of equations Augmented matrix Reduced matrix Reduced system of equation Solution Procedure for solving a system of linear equatio

28、ns 1. Echelon Form Definition 1.2.1: An (m n) matrix B is in echelon form if: All rows that consist entirely of zeros are grouped together at the bottom of the matrix. In every nonzero row, the first nonzero entry (counting from left to right) is a 1. 1. If the (i+1)-st row contains nonzero entries,

29、 then the first nonzero entry is in a column to the right of the first nonzero entry in the ith row. Definition 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 show

30、n, choose one of the following phrases to describe the matrix. The matrix is not in echelon form. The matrix is in echelon form, but not in reduced echelon form. 1. The matrix is in reduced echelon form. 143 012 001 A 100 110 231 B 0000 1000 0110 C 01000 32100 54321 D 0 0 1 E 1 0 0 F 001G100H 2. Sol

31、ving a linear system whose augmented matrix is in reduced echelon form 0000 7100 2010 3001 B Example 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. 1000 0

32、310 0101 C Solution: Matrix B is the augmented matrix for the following system: 7 2 3 3 2 1 x x x Therefore, the system has the unique solution x1=3, x2=-2, and x3=7. Matrix C is the augmented matrix for the following system: 1000 03 0 321 32 31 xxx xx xx Because no values for x1, x2, or x3 can sati

33、sfy the third equation, the system is inconsistent. 3. Recognizing an inconsistent system Theorem1.2.1: Let A|b be the augmented matrix for an (m n) linear system 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 syste

34、m of equations has no solution. That is , the system represented by A|b is inconsistent. 4. Solving a system of equations Step 1. Create the augmented matrix for the system. Step 2. Transform the matrix in Step 1 to reduced echelon form. Step 3. Decode the reduced matrix found in Step 2 to obtain it

35、s associated system of equations. Step 4. By examining the reduced system in Step 3, describe the solution set for the original system. 5. Reduction to echelon form Theorem1.2.2: Let B be an (m n) matrix. There is a unique (m n) matrix C such that: C is in reduced echelon form (1)C is row equivalent

36、 to B. Reduction 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 the 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 e

37、ntry in row one, multiply each entry in row one by 1/a. (4)Add appropriate multiples of row one to each of the remaining rows so that every entry below the 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 proces

38、s 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 multiples of each nonzero row to the rows above in order to zero all entries above the leading 1. Example3: Use elementary row operations to t

39、ransform the following matrix to reduced echelon form . 211761820 3324931230 91131000 4820000 A Exercise: . 1213345 2362210 231123 711111 A 612881063 1063 13522 28114342 54321 543 54321 54321 xxxxx xxx xxxxx xxxxx Example 4: Solve the following system of equations: Solution: transform the augmented

40、matrix to reduced echelon form. 612881063 1061300 1352121 28114342 612881063 1061300 1352121 1562221 R1+R2R2+R1 R4-3R1 000000 431000 210100 320021 Then matrix above represents the following system of equations 43 2 322 54 53 521 xx xx xxx 16102400 1061300 210100 1562221 R1+2R3 R4+2R3 862000 431000 2

41、10100 1142021 R1-2R2 R3+3R2 R4-4R2 Remark: 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 the system. For this example, x2 and x5 are independent variables and can be assigned values arbitrarily. The variables

42、x1,x3, and x4 are dependent variables, and their values are determined by the values assigned to x2 and x5. Particular solution. ) 1 ( 34 2 223 54 53 521 xx xx xxx Solving the preceding system, we find: Exercises : P27 28,30,49,53 Three people play a game in which there are always two winners and on

43、e 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? P 24 .example 1.31.3 Consistent Systems of Consistent Systems of Linear EquationsLine

44、ar Equations 1. Solution possibilities for a consistent linear system mnmnmm nn nn bxaxaxa bxaxaxa bxaxaxa 2211 22222121 11212111 )1( Our goal is to deduce as much information as possible about the solution set of system (1) without actually solving the system. Theorem1.3.1: Let the matrix C|d is in

45、 reduced echelon form. The 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. Theor

46、em1.3.4: Let r denote the number of nonzero rows in C|d. If the system represented by C|d is consistent, then r n. 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. 2. Homogeneous Systems The (m n) system o

47、f linear equations given in (2) is called a homogeneous system of linear equations: 0 )2(0 0 2211 2222121 1212111 nmnmm nn nn xaxaxa xaxaxa xaxaxa 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 i

48、n the solution of the system there are n-r variables that can be assigned arbitrary values. A homogeneous system is always consistent, because x1=x2= xn=0 is a solution to system (2). This solution is called the trivial solution(平凡解）平凡解） or zero solution, and any other solution is called a nontrivia

49、l solution. Theorem1.3.7: A homogeneous (m n) system of linear equations always has infinitely 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

50、not nonsingular. Theorem1.7.2: The (n n) matrix A=A1,A2,An is nonsingular if and only is A1,A2,An is 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 49,50 1.91

51、.9 Matrix inverses and Matrix inverses and their propertiestheir properties 1. The matrix inverse Definition1.9.1: Let A be an (n n) matrix. We say that A is invertible if we can find an (n n) matrix A-1 such that . 11 IAAAA The matrix A-1 is called an inverse for A. Example1: Let , 43 21 Afind its

52、inverse matrix. 2. Using inverses to solve systems of linear equations Ax=b x=A-1b AX=B X=A-1B where A is an (n n) matrix, B and X are (n m) matrices. 3. Existence of inverses Lemma: 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 a

53、n (n n) matrix. Then A has an inverse if and only if A is nonsingular. 4. Calculating the inverse Computation of A-1 Step1. Form the (n 2n) matrix A|I. Step2. Use elementary row operations to transform A|I to the form I|B. Step3. Reading form this final form, A-1=B. 101 3 2 3 1 1 3 4 3 1 2 100 010 001 Example1: 。 1 find, 311 021 211 Let AA solution： 100 010 001 311 021 211 ）（IA 101 011 001 100 230 211 101 213 203 100 030 011 101 3 2 3 1 1 203 100 010 011 1 22 3 1 1 101 3 2 3 1