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136_course_notes - Linear Algebra 1 Course Notes for MATH...

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Unformatted text preview: Linear Algebra 1 Course Notes for MATH 136 Edition 1.0 D. Wolczuk Copyright: D. Wolczuk, 1st Edition, 2011 Contents A Note to Students - READ THIS! . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Vectors in Euclidean Space 1 1.1 Vector Addition and Scalar Multiplication . . . . . . . . . . . . . . . 1 1.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Systems of Linear Equations 28 2.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 28 2.2 Solving Systems of Linear Equation . . . . . . . . . . . . . . . . . . . 32 3 Matrices and Linear Mappings 46 3.1 Operations on Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Special Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Operations on Linear Mappings . . . . . . . . . . . . . . . . . . . . . 75 4 Vector Spaces 78 4.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5 Inverses and Determinants 107 5.1 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4 Determinants and Systems of Equations . . . . . . . . . . . . . . . . 129 5.5 Area and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6 Diagonalization 138 6.1 Matrix of a Linear Mapping and Similar Matrices . . . . . . . . . . . 138 6.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 143 6.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4 Powers of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 ii A NOTE TO STUDENTS - READ THIS! Best Selling Novel: My Favourite Math, by Al G. Braw Welcome to Linear Algebra! In this course, you will be introduced to basic elements in linear algebra, including vectors, vector spaces, matrices, linear mappings, and solving systems of linear equa- tions. The strategy used in this course is to help you understand an example of a concept before generalizing it. For example, in Chapter 1, you will learn about vectors in R n , and then in Chapter 4, we will extend part of what we did with vectors in R n to the abstract setting of a vector space. Of course, the better you understand the particular example, the easier it will be for you to understand the more complicated concepts introduced later....
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This note was uploaded on 01/21/2012 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.

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136_course_notes - Linear Algebra 1 Course Notes for MATH...

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