Elementary Linear Algebra - K. R. MATTHEWS

Elementary Linear Algebra - K. R. MATTHEWS - ELEMENTARY...

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Unformatted text preview: ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at [email protected] All contents copyright c 1991 Keith R. Matthews Department of Mathematics University of Queensland All rights reserved 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498....
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Elementary Linear Algebra - K. R. MATTHEWS - ELEMENTARY...

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