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Unformatted text preview: APPLIED LINEAR ALGEBRA — THE LECTURE NOTES FOR MATH- 270 (2010)-Fall APPLIED LINEAR ALGEBRA JIAN-JUN XU Department of Mathematics and Statistics, McGill University Kluwer Academic Publishers Boston/Dordrecht/London Contents 1. SOLUTIONS OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 1 1 Introduction 1 2 Solutions for general ( m × n ) System of Linear Equations 1 2.1 (*)Elementary Row Operations 3 2.2 Gaussian Elimination 5 2.3 Gauss-Jordan Elimination 6 2.4 General Algorithm for Gaussian Elimination with Pivoting 7 2.5 Rank of Matrix 8 2.5.1 Existence and uniqueness of solution 8 2.6 Subspaces, Basis, and Dimension 9 2. MATRICES AND DETERMINANTS 13 1 Matrix Algebra 13 1.1 Matrix Addition 13 1.2 Scalar Multiplication of Matrix 13 1.3 Matrix Multiplication 13 1.3.1 demonstration of matrix multiplication 13 1.3.2 definition of matrix multiplication 15 1.4 Some Properties of Matrix Multiplication 15 2 Some Important ( n × n ) Matrices 16 2.1 Row Elementary Operation Matrices 17 3 Partitioning of Matrices 17 4 Transpose Matrix 19 5 The Inverse of a Square Matrix 20 v vi APPLIED LINEAR ALGEBRA 5.1 Basic Theorems on the Inverse of Matrix 20 5.2 Derivation of the Inverse of Matrix 21 5.2.1 Method (I): 21 5.2.2 Method (II): 22 6 The Matrix Exponential Function 22 6.1 The Basic properties of Matrix Exponential Function 22 6.2 Calculation of Function e At for Some Special Cases 22 6.2.1 For diagonal matrix 23 6.2.2 ( * ) For non-defective matrix 23 6.3 Differentiation of matrix exponential function 23 7 Determinant 24 7.1 Basic concepts and Definition of Determinant 25 7.2 Theorems of Determinant 27 7.3 Properties of Determinant 28 7.4 Minor, Co-factors, Adjoint 30 8 Cramer’s Rule 33 9 L-U Factorization 34 3. VECTOR SPACE: DEFINITIONS AND BASIC CONCEPTS 37 1 Introduction and Motivation 37 1.1 A brief Review of Geometric Vectors in 3D space 37 2 Definition of a Vector Space 39 3 Examples of Vector Spaces 40 4 Subspaces 41 5 Linear Combinations and Spanning Sets 42 6 Linear Dependence and Independence 43 7 Bases and Dimensions 43 4. INNER PRODUCT SPACES 49 1 Introduction 49 1.1 Definition of Inner Product in R n 50 1.1.1 Schwarz Inequality 50 1.1.2 Triangle Inequality 51 1.2 Basic Properties of the Inner Product in R n 51 2 Definition of a Real Inner Product Space 51 3 Definition of a Complex Inner Product Space 52 Contents vii 4 (*)The Gram-Schmidt Orthogonalization Procedure 54 5 Some Further concepts and notations 58 5.1 The Fourier coefficients of vector with respect to the orthogonomal set of basis 58 5.2 The projection of vector on a subspace 59 5.3 The orthogonal complement of a subset 59 5. LINEAR TRANSFORMATIONS 61 1 Introduction 61 2 Definitions 61 3 Matrix Representation of Linear Transformation 63 3.1 Some Special Linear Transformations and their Matrix Representations 64 4 The Kernel and Range of a Linear Transformation 69 5 Calculating the Range and Kernel with Matrix Representation 73 6 Algebra of Linear Transformation 76 6.1 Product of LT’s induced by matrices...
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book - APPLIED LINEAR ALGEBRA — THE LECTURE NOTES FOR...

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