Linear Algebra

Description

Linear Algebra is intended primarily as an undergraduate textbook but is written in such a way that it can also be a valuable resource for independent learning. The narrative of the book takes a matrix approach: the exposition is intertwined with matrices either as the main subject or as tools to explore the theory. Each chapter contains a description of its aims, a summary at the end of the chapter, exercises, and solutions. The reader is carefully guided through the theory and techniques presented which are outlined throughout in "How to…" text boxes. Common mistakes and pitfalls are also pointed out as one goes along. Features Written to be self-contained Ideal as a primary textbook for an undergraduate course in linear algebra Applications of the general theory which are of interest to disciplines outside of mathematics, such as engineering Cover Half Title Title Page Copyright Page Dedication Contents Preface Symbol Description Biography 1. Matrices 1.1. Real and Complex Matrices 1.2. Matrix Calculus 1.3. Matrix Inverses 1.4. Elementary Matrices 1.4.1. LU and LDU factorisations 1.5. Exercises 1.6. At a Glance 2. Determinant 2.1. Axiomatic Definition 2.2. Leibniz’s Formula 2.3. Laplace’s Formula 2.4. Exercises 2.5. At a Glance 3. Vector Spaces 3.1. Vector Spaces 3.2. Linear Independence 3.3. Bases and Dimension 3.3.1. Matrix spaces and spaces of polynomials 3.3.2. Existence and construction of bases 3.4. Null Space, Row Space, and Column Space 3.4.1. Ax = b 3.5. Sum and Intersection of Subspaces 3.6. Change of Basis 3.7. Exercises 3.8. At a Glance 4. Eigenvalues and Eigenvectors 4.1. Spectrum of a Matrix 4.2. Spectral Properties 4.3. Similarity and Diagonalisation 4.4. Jordan Canonical Form 4.4.1. Nilpotent matrices 4.4.2. Generalised eigenvectors 4.4.3. Jordan canonical form 4.5. Exercises 4.6. At a Glance 5. Linear Transformations 5.1. Linear Transformations 5.2. Matrix Representations 5.3. Null Space and Image 5.3.1. Linear transformations T : Kn → Kk 5.3.2. Linear transformations T : U → V 5.4. Isomorphisms and Rank-nullity Theorem 5.5. Composition and Invertibility 5.6. Change of Basis 5.7. Spectrum and Diagonalisation 5.8. Exercises 5.9. At a Glance 6. Inner Product Spaces 6.1. Real Inner Product Spaces 6.2. Complex Inner Product Spaces 6.3. Orthogonal Sets 6.3.1. Orthogonal complement 6.3.2. Orthogonal projections 6.3.3. Gram–Schmidt process 6.4. Orthogonal and Unitary Diagonalisation 6.5. Singular Value Decomposition 6.6. Affine Subspaces of Rn 6.7. Exercises 6.8. At a Glance 7. Special Matrices by Example 7.1. Least Squares Solutions 7.2. Markov Chains 7.2.1. Google matrix and PageRank 7.3. Population Dynamics 7.4. Graphs 7.5. Differential Equations 7.6. Exercises 7.7. At a Glance 8. Appendix 8.1. Uniqueness of Reduced Row Echelon Form 8.2. Uniqueness of Determinant 8.3. Direct Sum of Subspaces 9. Solutions 9.1. Solutions to Chapter 1 9.2. Solutions to Chapter 2 9.3. Solutions to Chapter 3 9.4. Solutions to Chapter 4 9.5. Solutions to Chapter 5 9.6. Solutions to Chapter 6 9.7. Solutions to Chapter 7 Bibliography Index