The Linear Algebra a Beginning Graduate Student Ought to Know

Portada
Springer Science & Business Media, 5 d’abr. 2007 - 436 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.

 

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Pàgines seleccionades

Continguts

Notation and terminology
2
Fields
5
Vector spaces over a field
17
Algebras over a field
33
Linear independence and dimension
49
Linear transformations
79
The endomorphism algebra of a vector space
99
Representation of linear transformations by matrices
117
Krylov subspaces
267
The dual space
285
Inner product spaces
299
Orthogonality
325
Selfadjoint Endomorphisms
349
Unitary and Normal endomorphisms
369
MoorePenrose pseudoinverses
389
Bilinear transformations and forms
399

The algebra of square matrices
131
Systems of linear equations
169
Determinants
199
Eigenvalues and eigenvectors
229
Summary of Notation
423
Index
427
Copyright

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