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

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Linear Algebra 1 Course Notes for MATH 136 Edition 1.0 D. Wolczuk Copyright: D. Wolczuk, 1st Edition, 2011

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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

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

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