Linear Algebra for the Young Mathematician (Pure and Applied Undergraduate Texts)

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 1 Title page 4 Preface 10 Part I . Vector spaces 14 Chapter 1. The basics 16 1.1. The vector space Fn 16 1.2. Linear combinations 20 1.3. Matrices and the equation Ax=b 24 1.4. The basic counting theorem 29 1.5. Matrices and linear transformations 34 1.6. Exercises 38 Chapter 2. Systems of linear equations 42 2.1. The geometry of linear systems 42 2.2. Solving systems of equations—setting up 49 2.3. Solving linear systems—echelon forms 53 2.4. Solving systems of equations—the reduction process 57 2.5. Drawing some consequences 63 2.6. Exercises 67 Chapter 3. Vector spaces 70 3.1. The notion of a vector space 70 3.2. Linear combinations 75 3.3. Bases and dimension 83 3.4. Subspaces 92 3.5. Affine subspaces and quotient vector spaces 104 3.6. Exercises 108 Chapter 4. Linear transformations 116 4.1. Linear transformations I 116 4.2. Matrix algebra 120 4.3. Linear transformations II 125 4.4. Matrix inversion 135 4.5. Looking back at calculus 141 4.6. Exercises 145 Chapter 5. More on vector spaces and linear transformations 152 5.1. Subspaces and linear transformations 152 5.2. Dimension counting and applications 155 5.3. Bases and coordinates: vectors 161 5.4. Bases and matrices: linear transformations 171 5.5. The dual of a vector space 180 5.6. The dual of a linear transformation 184 5.7. Exercises 190 Chapter 6. The determinant 208 6.1. Volume functions 208 6.2. Existence, uniqueness, and properties of the determinant 215 6.3. A formula for the determinant 219 6.4. Practical evaluation of determinants 224 6.5. The classical adjoint and Cramer’s rule 226 6.6. Jacobians 227 6.7. Exercises 229 Chapter 7. The structure of a linear transformation 234 7.1. Eigenvalues, eigenvectors, and generalized eigenvectors 235 7.2. Polynomials in cT 239 7.3. Application to differential equations 247 7.4. Diagonalizable linear transformations 252 7.5. Structural results 259 7.6. Exercises 266 Chapter 8. Jordan canonical form 272 8.1. Chains, Jordan blocks, and the (labelled) eigenstructure picture of cT 273 8.2. Proof that cT has a Jordan canonical form 276 8.3. An algorithm for Jordan canonical form and a Jordan basis 281 8.4. Application to systems of first-order differential equations 288 8.5. Further results 293 8.6. Exercises 296 Part II . Vector spaces with additional structure 302 Chapter 9. Forms on vector spaces 304 9.1. Forms in general 304 9.2. Usual types of forms 311 9.3. Classifying forms I 314 9.4. Classifying forms II 316 9.5. The adjoint of a linear transformation 323 9.6. Applications to algebra and calculus 330 9.7. Exercises 332 Chapter 10. Inner product spaces 338 10.1. Definition, examples, and basic properties 338 10.2. Subspaces, complements, and bases 344 10.3. Two applications: symmetric and Hermitian forms, and the singular value decomposition 352 10.4. Adjoints, normal linear transformations, and the spectral theorem 363 10.5. Exercises 372 Appendix A. Fields 380 A.1. The notion of a field 380 A.2. Fields as vector spaces 381 Appendix B. Polynomials 384 B.1. Statement of results 384 B.2. Proof of results 386 Appendix C. Normed vector spaces and questions of analysis 390 C.1. Spaces of sequences 390 C.2. Spaces of functions 392 Appendix D. A guide to further reading 396 Index 398 Back Cover 406 Provides a careful, thorough, and rigorous introduction to linear algebra. The book 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. __Linear Algebra for the Young Mathematician__