The Linear Algebra a Beginning Graduate Student Ought to KnowSpringer Science & Business Media, 31 de gen. 2004 - 406 pàgines Linear algebra is a living, active branch of mathematics which is central to almost all other areas of mathematics, both pure and applied, as well as computer science, the physical and social sciences, and engineering. It entails an extensive corpus of theoretical results as well as a large body of computational techniques. Unfortunately, in recent years the content of the linear algebra courses required to complete an undergraduate degree in mathematics has been depleted to the extent that they fail to provide a sufficient theoretical or computational background. Students are not only less able to formulate or even follow mathematical proofs, they are also less able to understand the mathematics of the numerical algorithms they need for applications. Certainly, the material presented in the average undergraduate linear algebra course is insufficient for graduate study. This book is intended to fill this gap by providing enough material "theoretical and computational" to allow the student to work independently or in advanced courses. The book is intended to be used in one of several possible ways: (1) as a self-study guide ; (2) as a textbook for a course in advanced linear algebra, either at the upper-class undergraduate level or at the first-year graduate level ; or (3) as a reference book. It is also designed to prepare a student for the linear algebra portion of prelim exams or PhD qualifying exams. The volume is self-contained to the extent that it does not assume any previous formal knowledge of linear algebra, though the reader is assumed to have been exposed, at least informally, to some basic ideas and techniques, such as the solution of a small system of linear equations over the real numbers. More importantly, it does assume a seriousness of purpose and a modicum of mathematical sophistication on the part of the reader. The book also contains over 1000 exercises, many of which are very challenging. |
Continguts
Notation and terminology | 1 |
Fields | 5 |
Vector spaces over a field | 17 |
Algebras over a field | 33 |
Linear dependence and dimension | 47 |
Linear transformations | 75 |
The endomorphism algebra of a vector space | 91 |
Representation of linear transformations by matrices | 97 |
Krylov subspaces | 235 |
The dual space | 253 |
Inner product spaces | 267 |
Orthogonality | 289 |
Selfadjoint Endomorphisms | 313 |
Unitary and Normal endomorphisms | 331 |
MoorePenrose pseudoinverses | 351 |
Bilinear transformations and forms | 361 |
The algebra of square matrices | 111 |
Systems of linear equations | 147 |
Determinants | 175 |
Eigenvalues and eigenvectors | 201 |
Summary of Notation | 381 |
Index | 385 |
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Frases i termes més freqüents
algebra algorithm associated assume Aut(V automorphism Bil(V bilinear form canonical basis characteristic polynomial columns complex numbers denote diagonal matrix dim(V dim(W dot product eigenspace eigenvalue eigenvector End(V endomorphism entries equal Example Exercise F and let F-algebra field and let field F Find finite dimension finitely-generated inner product function defined given Hom(V im(a inner product space integer and let isomorphism ker(a Let F linear equations linear functional linear transformation linearly independent linearly-independent subset mathematician matrix in Mnxn minimal polynomial Mkxn Mnxn F monic monomorphism Moreover nilpotent nonempty set nonsingular matrix nonzero orthogonal orthonormal basis positive definite positive integer Proof Proposition represented with respect satisfying the condition selfadjoint Show solution space of finite space over F spec(a subspace symmetric system of linear unique unital F-algebra unitary v₁ vector space vector space finitely W₁ W₂