THEORY OF EQUATIONS, MATRICES AND VECTOR CALCULUS MAT 4 B04
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1 THEORY OF EQUATIONS, MATRICES AND VECTOR CALCULUS MAT 4 B04 Core course of BSc Mathematics IV semester CUCBCSS 2014 Admn onwards UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION Calicut University PO, Malappuram, Kerala, India A
2 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION STUDY MATERIAL THEORY OF EQUATIONS, MATRICES AND VECTOR CALCULUS Core course of BSc Mathematics CUCBCSS 2014 Admission onwards Prepared by : Aboobacker P Assistant Professor Department of Mathematics W M O College, Muttil,Wayanad Scrutinized by: Dr.D.Jayaprasad, Principal, Sreekrishna College, Guruvayur Chairman, Board of Studies in Mathematics (UG) Lay out : Computer Section, SDE Reserved Page 2
3 CONTENTS Module 1 Theory of Equations Module 2 Rank Of A Matrix Module 3 System of Linear Homogeneous Equations Module 4 Lines and Planes in Space Page 3
4 Page 4
5 MODULE -1 Page 5
6 Page 6
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21 or x= y/k Page 21
22 To get equation with roots as reciprocals of original roots 3. To form an equation whose roots are less by h then the roots of a given equation. ( i.e., Diminishing the roots by h ) Page 22
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40 Module 2 RANK OF A MATRIX Page 40
41 Problem 3 :Prove that the rank of matrix every element of which is unity is 1 Solution; Since all elements are 1, square matrix of every order will have determinant 0, except the square matrix [1] of order 1. Problem 4: Show that no skew-symmetric matrix can be of rank 1. Solution: Let A be a skew-symmetric matrix. If A is zero matrix, then Page 41
42 Elementary Transformation and Equivalent Matrices Definition 17. An elementary transformation is an operation of any one of the following types: Definition. Equivalent Matrices: Two matrices are said to be equivalent if one can be obtained from the other by a nite number of elementary transformations. Exercise: Prove that equivalent matrices have same rank. Determination of rank using elementary transformations Theorem. Every m n matrix of rank r can by a sequence of elementary transformations be reduced to any one of the form Page 42
43 Definition. One of the forms Theorem. Page 43
44 Page 44
45 Row Reduced Echelon Forms A matrix is in echelon form when 1) Each row containing a non-zero number has the number 1 appearing in the rowʼs first nonzero column.(such an entry will be referred to as a leading one.) 2) The column numbers of the columns containing the first non-zero entries in each of the rows strictly increases from the first row to the last row. (Each leading one is to the right of any leading one above it.) 3) Any row which contains all zeros is below the rows which contain a non-zero entry. The three conditions above will ensure that the entries below the leading ones (in each row which contains a nonzero entry) are all zeros. A matrix is in reduced echelon form when: in addition to the three conditions for a matrix to be in echelon form, the entries above the leading ones (in each row which contains a non-zero entry) are all zeroʼs. Note : If a matrix is in Reduced Row Echelon Form then it must also be in Echelon form. Page 45
46 Solution Interchange first and second rows Page 46
47 To find reduced echelon form The matrix I is in reduced echelon form. Page 47
48 Page 48
49 MODULE III Page 49
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54 Characteristic Values And Characteristic Vectors Of A Martix Deinition : Page 54
55 Page 55
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58 Cayley Hamilton Theorem Page 58
59 UNIT IV LINES AND PLANES IN SPACE Page 59
60 ANALYTIC GEOMETRY IN SPACE Page 60
61 ANALYTIC GEOMETRY IN SPACE Page 61
62 Page 62
63 Angle Between two Vectors If is the angle between two vectors a and b, then Page 63
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67 In terms of determinant Page 67
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71 This is the required equation. Page 71
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77 1. Page 77
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Math "Matrix Approach to Solving Systems" Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25
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