Linear Algebra for the Young Mathematician

Description

Linear Algebra for the Young Mathematician is a careful, thorough, and rigorous introduction to linear algebra. It adopts a conceptual point of view, focusing on the notions of vector spaces and linear transformations, and it takes pains to provide proofs that bring out the essential ideas of the subject. It begins at the beginning, assuming no prior knowledge of the subject, but goes quite far, and it includes many topics not usually treated in introductory linear algebra texts, such as Jordan canonical form and the spectral theorem. While it concentrates on the finite-dimensional case, it treats the infinite-dimensional case as well. The book illustrates the centrality of linear algebra by providing numerous examples of its application within mathematics. It contains a wide variety of both conceptual and computational exercises at all levels, from the relatively straightforward to the quite challenging. br>Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true. Cover Title page Preface Part I . Vector spaces Chapter 1. The basics 1.1. The vector space Fⁿ 1.2. Linear combinations 1.3. Matrices and the equation 𝐴𝑥=𝑏 1.4. The basic counting theorem 1.5. Matrices and linear transformations 1.6. Exercises Chapter 2. Systems of linear equations 2.1. The geometry of linear systems 2.2. Solving systems of equations—setting up 2.3. Solving linear systems—echelon forms 2.4. Solving systems of equations—the reduction process 2.5. Drawing some consequences 2.6. Exercises Chapter 3. Vector spaces 3.1. The notion of a vector space 3.2. Linear combinations 3.3. Bases and dimension 3.4. Subspaces 3.5. Affine subspaces and quotient vector spaces 3.6. Exercises Chapter 4. Linear transformations 4.1. Linear transformations I 4.2. Matrix algebra 4.3. Linear transformations II 4.4. Matrix inversion 4.5. Looking back at calculus 4.6. Exercises Chapter 5. More on vector spaces and linear transformations 5.1. Subspaces and linear transformations 5.2. Dimension counting and applications 5.3. Bases and coordinates: vectors 5.4. Bases and matrices: linear transformations 5.5. The dual of a vector space 5.6. The dual of a linear transformation 5.7. Exercises Chapter 6. The determinant 6.1. Volume functions 6.2. Existence, uniqueness, and properties of the determinant 6.3. A formula for the determinant 6.4. Practical evaluation of determinants 6.5. The classical adjoint and Cramer’s rule 6.6. Jacobians 6.7. Exercises Chapter 7. The structure of a linear transformation 7.1. Eigenvalues, eigenvectors, and generalized eigenvectors 7.2. Polynomials in cT 7.3. Application to differential equations 7.4. Diagonalizable linear transformations 7.5. Structural results 7.6. Exercises Chapter 8. Jordan canonical form 8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT 8.2. Proof that cT has a Jordan canonical form 8.3. An algorithm for Jordan canonical form and a Jordan basis 8.4. Application to systems of first-order differential equations 8.5. Further results 8.6. Exercises Part II . Vector spaces with additional structure Chapter 9. Forms on vector spaces 9.1. Forms in general 9.2. Usual types of forms 9.3. Classifying forms I 9.4. Classifying forms II 9.5. The adjoint of a linear transformation 9.6. Applications to algebra and calculus 9.7. Exercises Chapter 10. Inner product spaces 10.1. Definition, examples, and basic properties 10.2. Subspaces, complements, and bases 10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition 10.4. Adjoints, normal linear transformations, and the spectral theorem 10.5. Exercises Appendix A. Fields A.1. The notion of a field A.2. Fields as vector spaces Appendix B. Polynomials B.1. Statement of results B.2. Proof of results Appendix C. Normed vector spaces and questions of analysis C.1. Spaces of sequences C.2. Spaces of functions Appendix D. A guide to further reading Index Back Cover