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Elementary Linear Algebra( K.R. Matthews)

# Elementary Linear Algebra( K.R. Matthews) - ELEMENTARY...

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ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at [email protected]

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Contents 1 LINEAR EQUATIONS 1 1.1 Introduction to linear equations . . . . . . . . . . . . . . . . . 1 1.2 Solving linear equations . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Gauss–Jordan algorithm . . . . . . . . . . . . . . . . . . 8 1.4 Systematic solution of linear systems. . . . . . . . . . . . . . 9 1.5 Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . 16 1.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 MATRICES 23 2.1 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Linear transformations . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Non–singular matrices . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Least squares solution of equations . . . . . . . . . . . . . . . 47 2.7 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 SUBSPACES 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Subspaces of F n . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Basis of a subspace . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5 Rank and nullity of a matrix . . . . . . . . . . . . . . . . . . 64 3.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 DETERMINANTS 71 4.1 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 i
5 COMPLEX NUMBERS 89 5.1 Constructing the complex numbers . . . . . . . . . . . . . . . 89 5.2 Calculating with complex numbers . . . . . . . . . . . . . . . 91 5.3 Geometric representation of C . . . . . . . . . . . . . . . . . . 95 5.4 Complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Modulus of a complex number . . . . . . . . . . . . . . . . . 99 5.6 Argument of a complex number . . . . . . . . . . . . . . . . . 103 5.7 De Moivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . 107 5.8 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6 EIGENVALUES AND EIGENVECTORS 115 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Definitions and examples . . . . . . . . . . . . . . . . . . . . . 118 6.3 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7 Identifying second degree equations 129 7.1 The eigenvalue method . . . . . . . . . . . . . . . . . . . . . . 129 7.2 A classification algorithm . . . . . . . . . . . . . . . . . . . . 141 7.3 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8 THREE–DIMENSIONAL GEOMETRY 149 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2 Three–dimensional space . . . . . . . . . . . . . . . . . . . . . 154 8.3 Dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.5 The angle between two vectors . . . . . . . . . . . . . . . . . 166 8.6 The cross–product of two vectors . . . . . . . . . . . . . . . . 172 8.7 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.8 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9 FURTHER READING 189 ii

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List of Figures 1.1 Gauss–Jordan algorithm . . . . . . . . . . . . . . . . . . . . . 10 2.1 Reflection in a line . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Projection on a line . . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Area of triangle OPQ . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 Complex addition and subtraction . . . . . . . . . . . . . . . 96 5.2 Complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3 Modulus of a complex number . . . . . . . . . . . . . . . . . 99 5.4 Apollonius circles . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5 Argument of a complex number . . . . . . . . . . . . . . . . . 104 5.6 Argument examples . . . . . . . . . . . . . . . . . . . . . . . 105 5.7 The n th roots of unity . . . . . . . . . . . . . . . . . . . . . . . 108 5.8 The roots of z n = a . . . . . . . . . . . . . . . . . . . . . . . . 109 6.1 Rotating the axes . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.1 An ellipse example . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 ellipse: standard form . . . . . . . . . . . . . . . . . . . . . . 137 7.3 hyperbola: standard forms . . . . . . . . . . . . . . . . . . . . 138 7.4 parabola: standard forms (i) and (ii) . . . . . . . . . . . . . . 138 7.5 parabola: standard forms (iii) and (iv) . . . . . . . . . . . . . 139 7.6 1st parabola example . . . . . . . . . . . . . . . . . . . . . . . 140 7.7 2nd parabola example . . . . . . . . . . . . . . . . . . . . . . 141 8.1 Equality and addition of vectors . . . . . . . . . . . . . . . . 150 8.2 Scalar multiplication of vectors . . . . . . . . . . . . . . . . . . 151 8.3 Representation of three–dimensional space . . . . . . . . . . . 155 8.4 The vector AB . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.5 The negative of a vector . . . . . . . . . . . . . . . . . . . . . . 157 iii
1 8.6 (a) Equality of vectors; (b) Addition and subtraction of vectors.157 8.7 Position vector as a linear combination of i , j and k . . . . . . 158 8.8 Representation of a line . . . . . . . . . . . . . . . . . . . . . . 162 8.9 The line AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.10 The cosine rule for a triangle . . . . . . . . . . . . . . . . . . . 167 8.11 Pythagoras’ theorem for a right–angled triangle. . . . . . . . 168 8.12 Distance from a point to a line . . . . . . . . . . . . . . . . . . 169 8.13 Projecting a segment onto a line . . . . . . . . . . . . . . . . . 171 8.14 The vector cross–product . . . . . . . . . . . . . . . . . . . . . 174 8.15 Vector equation for the plane ABC . . . . . . . . . . . . . . . 177 8.16 Normal equation of the plane ABC . . . . . . . . . . . . . . . 178 8.17 The plane ax + by + cz = d . . . . . . . . . . . . . . . . . . . . 179 8.18 Line of intersection of two planes . . . . . . . . . . . . . . . . . 182 8.19 Distance from a point to the plane ax + by + cz = d . . . . . . 184

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2
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1 , x 2 , · · · , x n is an equation of the form a 1 x 1 + a 2 x 2 + · · · + a n x n = b, where a 1 , a 2 , . . . , a n , b are given real numbers.

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