CHAPTER-2-SOLVING-LINEAR-ALGERBRA111.pptx

Chapter 2 Solving Linear Algebra

1. Chapter 2: Solving Linear System

2. Learning Objectives At the end of the chapter, we will be able to: 1. Find row echelon form and reduced row echelon form; 2. Solve linear systems using gauss jordan reduction and gaussian elimination

3. Learning Objectives 3. Find the inverse of matrix 4. Determine the equivalent matrices 5. D iscuss the variant of gaussian elimination which is LU-Factorization.

4. Table of contents Row Echelon Form 2.1 Solving Linear Systems 2.2 Elementary Matrices; Finding   2.3 Equivalent Matrices 2.4 2.5 LU Factorization

5. ENCHELON FORM OF A MATRIX 2.1

6. An m x n matrix A is said to be in reduced row echelon from if it satisfies the following properties: All zero rows if there are any appear at the bottom of the matrix. The first nonzero entry from the left of a nonzero row is a 1. This entry is called a leading one of its rows. For each nonzero row, the leading one appears to the right and below any leading ones in preceding rows. If a column contains a leading one, then all other entries in that column are zero.

7. An m x n matrix satisfying properties (a) (b) and (c) is said to be in row echelon form.   EXAMPLE:

8. An elementary row (column) operation on a Matrix A is any one of the following operations: Type I: Interchange any two rows (column). Ex. R1↔R3 Type II: Multiply row (column) by nonzero number. Ex. -3R2→R2 Type III: Add and multiply of one row (column) to another. Ex. -3R2 + R4→R4

9. Example 1 Find a row echelon form of each of the given matrices record the row operations you perform using the notation for elementary row operations.

10. SOLUTION:

11. Example 2 Find the reduced row echelon form of each of the given matrices. Record the row operations you perform, using the notation for elementary row operations.

12. SOLUTION:

13. Example 3 Find a row echelon form and reduced row echelon form of the given matrices. Record the row operations you perform using the notation for elementary row operations.

14. SOLUTION:

15. EXERCISES Find the reduced row echelon form of each of the given matrices record the row operation you perform, using the notation for elementary row operations. 2. Find the row echelon form of each of the given matrices. Record the row operation you perform, using the notation for elementary row operations.

16. EXERCISES 3 . Find the reduce row echelon form of each of the given matrices. Record the row operation you perform, using the notation for elementary row operations.

17. SOLVING LINEAR SYSTEM 2.2

18. Linear equations are equations of the first order. These equations are defined for lines in the coordinate system. An equation for a straight line is called a linear equation. The general representation of the straight-line equation is y= mx+b , where m is the slope of the line and b is the y-intercept. Linear equations are those equations that are of the first order. These equations are defined for lines in the coordinate system.

19. The echelon forms are more efficiently in determining the solution of a linear system compared with the elimination method. Using the augmented matrix of a linear system together with an echelon form. we develop two methods for solving a system of m linear equations in n unknow. These methods take the augmented matrix of the linear system, perform elementary row operations on it, and obtain a new matrix that represents an equivalent linear system. The important point is that the latter linear system can be solved more easily.

20. Represents the augmented matrix of a linear system. The n the solution is quickly. found from the corresponding equations

21. The task of this section is to manipulate the augmented matrix representing a given linear system into a form from which the solution can be found more easily. We now apply row operations to the solution of linear systems.

22. Theorem 2.3 Let Ax = b and Cx = d be two linear systems each of m equations in n unknowns. If the augmented matrices [A ⁞ b] and [C ⁞ d] arc row equivalent, then the linear systems are equivalent; that is. they have the same solutions. Proof This follows from the definition of row equivalence and from the fact that the three elementary row operations on the augmented matrix are the three manipulations on linear systems which yield equivalent linear systems. We also note that if one system has no solution, then the other system has no solution.

23. Recall from Section 1.1 that the linear system of the form: is called a homogeneous system. We can also write (1) in matrix form as

24. We observe that we have developed the essential features of two very straight-forward methods for solving linear systems. The idea consists of starting with the linear system Ax = b, then obtaining a partitioned matrix [C ⁞ d] in either row echelon form or reduced row echelon form that is row equivalent to the augmented matrix [A ⁞ b].

25. The method where [C ⁞d] is in row echelon form is called Gaussian elimination, the method where [C ⁞d] is in reduced row echelon form is called Gauss'- Jordan reduction. Strictly speaking, the original Gauss-Jordan reduction was more along the lines described in the preceding Remark. The version presented in this book is more efficient. In actual practice, neither Gaussian elimination nor Gauss-Jordan reduction is used as much as the method involving the LU-factorization of A that is discussed in Section 2.5. However. Gaussian elimination and Gauss-Jordan reduction are fine for small problems, and we use the latter heavily in this book.

26. Gaussian elimination consists of two steps: Step 1. The transformation of the augmented matrix [ A ⁞ b] to the matrix [ C ⁞ d] in row echelon form using elementary row operations. Step 2. Solution of the linear system corresponding to the augmented matrix [ C ⁞ d] using back substitution for the case in which A ~ n x n. and the linear system Ax = h has a unique solution, the matrix [ C ⁞ d] has the following form.

27. Example 1. The linear system has the augmented matrix

28. Transforming this matrix to row echelon form, we obtain (verify) Using back substitution, we now have Thus, the solution is x = 2, y = -1, z = 3, which is unique.

29. 1. Consider the linear system. x + y +2z= - 1 x – 2y + z = -5 3x+ y + z= 3. (a) Find all solutions, if any exist. by using the Gaussian elimination method. (b) Find all solutions. if any exist. by using the Gauss10rdan reduction method

30. 2. Find an equation relating (a. b. and c so that the linear system x + 2-3z = a 2x + 3y + 3z = b 5x + 9y – 6 = c is consistent for any values of (a, b, and c that satisfy that equation.

31. 3. Solve the linear system using the row echelon form. Record the row operation you perform, using the notation for elementary row operations. x + y + 2z + 3w = 13 x - 2y + z + w = 8 3x+ y + z- w=1

32. ELEMENTARY MATRICES  FINDING A^-1  2.3

33.   Definition:  An n*n elementary matrix of type I, type II, or type III is a matrix obtained from the identity matrix ln by performing a single elementary or elementary column operation of type I, type II, or type III respectively. 

34. Theorem of ELEMENTARY MATRICES  Theorem 2.5    Let A be an m*n matrix, and let an elementary row(column) operation of type I, type II or type III be performed on A to yield  matrix B. Let E be the elementary matrix obtained from lm (ln) by performing the same elementary row operation as was performed on A. Then B=EA(B=AE).  Example: We can readily verify that B= EA

35. Theorem 2.6       If A and B are m*n matrices, then A is row equivalent to B if and only if there exist elementary matrices E1,E2,…, Ek  such that B=  Ek  Ek-1… E2 E1A  (B=AE,…E8-1  Ek ).  Theorem 2.7       An elementary matrix E is nonsingular l, and its inverse is an elementary matrix of the same type. 

36. Lemma 2.1     Let A be an n*n matrix and let the homogeneous system Ax=0 have only the  trival  solution x=0. Then A is row equivalent to ln. ( That is the reduced tow echelon form of A is ln).  Theorem 2.8      A nonsingular if and only if A is a product of elementary matrices.  Corollary 2.2       A is nonsingular if and only if A is row equivalent to ln. ( That is the reduced row echelon form of A is ln). 

37. Theorem 2.9      The homogeneous system of n linear equations in n unknown Ax=0 has  nontrival  solution if and only if A is singular. ( That is, the reduced row echelon form of A is not equal to ln).  Example: Consider that homogeneous system Ax =0; that

38. The reduced row echelon form of the augmented Matrix is (Verify), so a solution is x= -2r y= r Where r is any real number. Thus the homogeneous system has a nontrivial solution, and A is singular.

39. Theorem 2.10    An n*n matrix A is singular if and only if A is row equivalent to matrix B that has a row of zeros ( That is, the reduced row echelon form of A has a row of zeros).  Theorem 2.11       If A and B are n*n matrices such that AB= lm , and BA=ln. Thus B A^-1 

40. Following are an examples of Elementary Matrices:

41. EXERCISES 1. Find the inverse of 2. Find the inverse of

42. EXERCISES 3. Determine if the following homogeneous system has a Nontrivial Solution. w+2x+3y+2z= 0 w+3x+5y+5z= 0 2w+4x+7y+X= 0 -w-2x-6y+7z= 0

43. EQUIVALENT MATRICES 2.4

44. We have thus far considered A to be row (column) equivalent to B if B results from A by finite sequence of elementary row (column) operations. A natural extension of this idea is that of considering B to arise from A by finite sequence of elementary row (column) operations. This leads to the notion of equivalence of matrices.

45. If A and B are two m x n matrices, then A is equivalent to B if we obtain B from A by a finite sequence of elementary row (column) operations.

46. I In the case of row (column) equivalent we can show that : A. Every matrix is equivalent to itself B. If B is equivalent to A, then A is equivalent to B C. If C is equivalent to B, and B is equivalent to A, then C is equivalent to A 

47. Let A = Applying elementary row operations, we obtain the following: B = A 2 r 3 + r 2 → r 2 = C= Br 2 ⇔ r 3 =  

48. D= C 2 r 1 → r 1 = So, we show that Matrices A,B,C and D are equivalent by applying the elementary row operations.. Then, we can also show that if two matrices are row equivalent, then they are equivalent.  

49. Let A and I be 2x2 matrices defined as follows : A= B= Prove that matrix A is equivalent I  

50. A = cr 1 – r 2 → r 2 = 1/ cb -d r 2 = br 2 – r 1 → r 1 = = I We can see that we obtain matrix B from A by finite sequence of elementary row (column) operations.  

51. Given that the following matrices are equivalent, find the value of x, y, z. Then, show the equivalent matrices. A = b =  

52. Solutions: x +3 = 6 y = 1 z – 3 = 4 x = 3 z = 7 Substitute the value of x, y, and z, we obtain A = B =  

53. 1 . Solve for x, y and z in the Matrix A and B, provided that they are equivalent A= B= 2. Given matrix T and Matrix Y, calculate the value of x and y T = S =   EXERCISES

54. EXERCISES 3. Solve for all variables of the A and B matrix. Then, prove that they are equivalent.   A = B =  

55. LU-Factorization 2.5

56. Example 1:

57. Submitted by Group 2 Dannazen E. Gullon Erica F. Palisoc Jhon Lloyd R. Palisoc Jessiel P. Quinto Elizabeth P. Soriano Christine P. Tejada Melanie E. Ventura

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