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Unformatted text preview: Linear Algebra Notes Brad Lackey DEPT OF MATHEMATICS, UNIV OF HULL, HULL HU6 7RX, UK Email address : [email protected] Contents Preliminaries 1 1. Arithmetic 1 2. Algebra 3 3. Geometry 3 4. Calculus 4 5. Logic and Set Theory 4 Part I. Applicable Linear Algebra 5 Chapter 1. Vectors and Matrices 6 1. Vectors in two dimensions 6 2. Vector in more than two dimensions 8 3. Matrices 8 4. Determinants 13 Chapter 2. Linear Systems 15 1. Substitution 15 2. Elimination 16 3. Row reduction 17 4. Reduced rowechelon form 18 5. Solution spaces of linear systems 20 6. Matrix inverses 21 7. Determinants revisited 23 Chapter 3. Scalar product 24 1. Properties of the scalar product 24 2. Orthogonality 24 3. Projections 25 4. Scalar products over C 26 Chapter 4. Eigenvalues and Eigenvectors 27 1. Eigenvalues and Eigenvectors 27 2. Diagonalization 28 Part II. Abstract Linear Algebra 33 Chapter 5. Vector spaces 34 1. Definitions and Examples 34 2. Linear Independence 35 3. Bases 38 iii CONTENTS iv Chapter 6. Inner product spaces 44 1. Definitions and examples 44 2. Orthonormal Bases 46 3. Projections 49 Chapter 7. Linear transformations 51 1. Definitions and Examples 51 2. Matrix of a Linear Transformation 52 3. Kernels and Images 55 4. Composition 60 Chapter 8. Jordan canonical form 64 1. Similarity (and change of basis) 64 Preliminaries Any University course in mathematics requires experience in secondary school topics. Although many lecturers would say that most skills required can be learned during the first year of University, this is just not the case. A student of mathematics will need a broad knowledge of basic facts and techniques to succeed in Universitylevel courses. This chapter is not designed to enhance, and certainly not to replace, preUniversity studies in mathematics. Yet, we must consider the paradoxical situation of knowing what is required for a course without knowing the course itself. It is for this reason we present some general topics which a student will find necessary to succeed at linear algebra. The following five subjects are quite extensive on their own, and we do not pretend to be comprehensive. However, if all the facts presented in the chapter are familiar, then one should be confident to proceed. We should emphasize that the material presented here is background for the following chapters, and it is comprehensive in this sense. Topics within the sections on arithmetic and algebra are naturally needed throughout the course, but the material in the section on geometry, calculus, and logic are not altogether mandatory. To truly master linear algebra, one must borrow from these last three subjects; yet, one can progress far without expertise in them....
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This note was uploaded on 02/24/2012 for the course MATH 513 taught by Professor Igordolgachev during the Winter '09 term at University of MichiganDearborn.
 Winter '09
 IgorDolgachev
 Calculus, Linear Algebra, Algebra, Geometry, Set Theory

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