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Preface This module is developed to meet the specific needs of students who may feel that they have deficiency in linear algebra and those students who have completed an undergraduate course in linear algebra. Each chapter begins with the learning objectives and pertinent definitions and theorems. All the illustrative examples and answers to the self-assessment quiz are fully explained to ensure complete understanding. The self-assessment quiz can provide students with ample opportunity for practice and can help them evaluate their progress. This self-learning module aims to provide students enrolled in the Open University of the Central Luzon State University a thorough understanding of the fundamental concepts of linear algebra and matrix theory through distance learning. It consists of seven chapters: Chapter 1 deals with matrix operations and their properties; Chapter 2 covers solutions of systems of linear equations using the Gauss-Jordan elimination method, determinants and its properties. Vectors in ℝ𝑛 , vector spaces and spanning sets are presented in Chapter 3. Chapter 4 deals with linear independence, basis and dimension of a matrix; Chapter 5 covers linear transformations and matrices. Eigenvalues, eigenvectors and diagonalization are discussed in Chapter 6 and inner product spaces, GramSchmidth orthogonalization process and diagonalization of symmetric matrix are discussed in Chapter 7. The author would like to thank those who have shared in the successful completion of this module. Dr. Flor Amor B. Monta, Director of Open University for the support, trust and confidence he gave to the author; Dr. Ruben C. Sevilleja, President of Central Luzon State University, for his continuous support and commitment to improve mathematics education; and, above all, to the Almighty God, for His bountiful blessings and graces. The Author Table of Contents Preface Chapter 1 Chapter 2 Chapter 3 Matrices Over a Field F Introduction Objectives Definition of a Field Definition of a Matrix Matrix Operations Properties of Matrix Addition Properties of Matrix Multiplication Properties of Scalar Multiplication Transpose of a Matrix Special Types of Square Matrices 1 1 1 2 3 7 9 11 12 14 Linear Equations and Matrices Introduction Objectives Solutions of Systems of Linear Equations Elementary Row Operations The Gauss-Jordan Method Homogeneous Systems The Inverse of a Matrix Linear Systems and Inverses Determinants Properties of Determinants Minors and Cofactor Expansion 19 19 19 23 27 32 35 41 46 49 53 Vector Spaces Over a Field Introduction Objectives Vectors in the Plane Vector Operations 𝒏-Vectors Vector Spaces and Subspaces Linear Combinations and Spanning Sets 64 64 64 65 65 67 76 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Linear Independence Introduction Objectives Definition of Linear Independence Basis and Dimension Rank of a Matrix Linear Transformations and Matrices Introduction Objectives Linear Transformations The Kernel and Range of a Linear Transformation The Matrix of a Linear Transformation Eigenvalues and Eigenvectors Introduction Objectives Characteristic Polynomial Hamilton-Cayley Theorem Eigenvalues, Eigenvectors, and Eigenspaces Diagonalization Inner Product Spaces Introduction Objectives Inner Product in ℝ𝑛 Orthonormal Bases in ℝ𝑛 Gram-Schmidt Orthogonalization Process Diagonalization of Symmetric Matrix 82 82 82 86 94 100 100 100 105 113 117 117 117 118 121 128 131 131 131 134 134 139