Linear algebra : concepts and methods

Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarize the material to help students consolidate their learning as they progress through the book Content: Preliminaries : before we begin -- Sets and set notation -- Numbers -- Mathematical terminology -- Basic algebra -- Trigonometry -- A little bit of logic -- 1. Matrices and vectors -- what is a matrix? -- Matrix addition and scalar multiplication -- Matrix multiplication -- Matrix algebra -- Matrix inverses -- Powers of a matrix -- The transpose and symmetric matrices -- Vectors -- Developing geometric insight -- Lines -- Planes -- Lines and hyperplanes -- 2. Systems of linear equations -- Row operations -- Gaussian elimination -- Homogeneous systems and null space -- 3. Matrix inversion and determinants -- Matrix inverse using row operations -- Determinants -- Matrix inverse using cofactors -- Leontief input-output analysis -- 4. Rank, range and linear equations -- Rank of a matrix -- Rank and systems of linear equations -- Range -- 5. Vector spaces -- Subspaces -- Linear span -- 6. Linear independence, bases and dimension -- Linear independence -- Bases -- Coordinates -- dimension -- Basis and dimension -- 7. Linear transformations and change of basis -- Range and null space -- Coordinate change -- Change of basis and similarity -- 8. Diagonalisation -- Eigenvalues and eigenvectors -- Diagonalisation of a square matrix -- 9. Applications of diagonalisation -- Powers of matrices -- Systems of difference equations -- Linear systems of differential equations -- 10. Inner products and orthogonality -- Orthogonal matrices -- Gram-Schmidt orthonormalisation process -- 11. Orthogonal diagonalisation and its applications -- Orthogonal diagonalisation of symmetric matrices -- Quadratic forms -- 12. Direct sums and projections -- Direct sum of two subspaces --Orthogonal complements -- Projections -- Characterising projections and orthogonal projections -- Orthogonal projection onto the range of a matrix -- Minimising the distance to a subspace -- Fitting function to data : least squares approximation -- 13. Complex matrices and vector spaces -- Complex numbers -- Complex vector spaces -- Complex matrices -- Complex inner product spaces -- Hermitian conjugates -- Unitary diagonalisation and normal matrices -- Spectral decomposition. Abstract: This thorough, yet concise, treatment of all the key topics also gives students a sound understanding of the underlying theory. Read more... "Any student studying linear algebra will welcome this textbook, which provides a thorough, yet concise, treatment of key topics in university linear algebra courses. Blending practice and theory, the book enables students to practice and master the standard methods as well as understand how they actually work. At every stage the authors take care to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses only on the fundamental topics. Hundreds of examples and exercises, including solutions, give students plenty of hands-on practice End-of-chapter sections summarize material to help students consolidate their learning Ideal as a course text and for self-study Instructors can use the many examples and exercises to supplement their own assignments Both authors have extensive experience of undergraduate teaching and of preparation of distance learning materials"-- Provided by publisher "Any student studying linear algebra will welcome this textbook, which provides a thorough, yet concise, treatment of key topics in university linear algebra courses. Blending practice and theory, the book enables students to practice and master the standard methods as well as understand how they actually work. At every stage the authors take care to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses only on the fundamental topics. Hundreds of examples and exercises, including solutions, give students plenty of hands-on practice End-of-chapter sections summarise material to help students consolidate their learning Ideal as a course text and for self-study Instructors can use the many examples and exercises to supplement their own assignments Both authors have extensive experience of undergraduate teaching and of preparation of distance learning materials"-- Provided by publisher Any student studying linear algebra will welcome this textbook, which provides a thorough, yet concise, treatment of key topics in university linear algebra courses. The authors, who have extensive teaching experience, provide hundreds of examples and exercises with a complete list of solutions, to enable students to practise and master the standard methods. Crucially, the authors also give clear explanations of how the methods really work, so that readers can gain a sound understanding of the underlying theory. End-of-chapter sections summarise the material to help students consolidate their learning as they progress through the book. At every stage the authors take care to ensure that the discussion is no more complicated or abstract than it needs to be and focuses only on the fundamental topics. Instructors can draw on the many examples and exercises to supplement their own assignments. Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarise the material to help students consolidate their learning as they progress through the book. Machine generated contents note: Preface; Preliminaries: before we begin; 1. Matrices and vectors; 2. Systems of linear equations; 3. Matrix inversion and determinants; 4. Rank, range and linear equations; 5. Vector spaces; 6. Linear independence, bases and dimension; 7. Linear transformations and change of basis; 8. Diagonalisation; 9. Applications of diagonalisation; 10. Inner products and orthogonality; 11. Orthogonal diagonalisation and its applications; 12. Direct sums and projections; 13. Complex matrices and vector spaces; 14. Comments on exercises; Index. This thorough, yet concise, textbook covers key topics in first- and second-year university courses and includes many examples and exercises with solutions to help students practise and master the relevant methods. Crucially, it fully develops the underlying theory so that students can understand how these methods really work.