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Matrix algebra for beginners, Part I matrices, determinants, inverses Jeremy Gunawardena Department of Systems Biology Harvard Medical School 200 Longwood Avenue, Cambridge, MA 02115, USA jeremy@hms.harvard.edu 3 January 2006 Contents 1 Introduction 1 2 Systems of linear equations 1 3 Matrices and matrix multiplication 2 4 Matrices and complex numbers 5 5 Can we use matrices to solve linear equations? 6 6 Determinants and the inverse matrix 7 7 Solving systems of linear equations 9 8 Properties of determinants 10 9 Gaussian elimination 11 1 1 Introduction This is a Part I of an introduction to the matrix algebra needed for the Harvard Systems Biology 101 graduate course. Molecular systems are inherently many dimensional—there are usually many molecular players in any biological system—and linear algebra is a fundamental tool for thinking about many dimensional systems. It is also widely used in other areas of biology and science. I will describe the main concepts needed for the course—determinants, matrix inverses, eigenvalues and eigenvectors—and try to explain where the concepts come from, why they are important and how they are used. If you know some of the material already, you may find the treatment here quite slow. There are mostly no proofs but there are worked examples in low dimensions. New concepts appear in italics when they are introduced or defined and there is an index of important items at the end. There are many textbooks on mat ...
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