id: 02052178
dt: b
an: 2014b.00592
au: Golan, Jonathan S.
ti: The linear algebra. A beginning graduate student ought to know.
so: Kluwer Texts in the Mathematical Sciences 27. Dordrecht: Kluwer Academic
Publishers (ISBN 1-4020-1824-X/hbk). x, 406~p. (2004).
py: 2004
pu: Dordrecht: Kluwer Academic Publishers
la: EN
cc: H65
ut: linear algebra; textbook; vector spaces; linear dependence; linear
transformations; endomorphism algebra; systems of linear equations;
determinants; eigenvalues; eigenvectors; Krylov subspaces; dual space;
inner product spaces; orthogonality; selfadjoint endomorphisms;
Moore-Penrose pseudoinverses; bilinear transformations; exercises
ci:
li:
ab: In recent years the trend has been for linear algebra courses and texts to
become less abstract, more informal and even shallow, in order to match
the requirements of students who come increasingly from other
disciplines and have only a rudimentary knowledge of mathematics.
Furthermore, the author of this book deplores the fact that at many
universities these are the only linear algebra courses available to
students wishing to complete an undergraduate degree in mathematics. In
the author’s words: ‘Students are not only less able to formulate
or even follow mathematical proofs, they are also less able to
understand the underlying mathematics of the numerical algorithms they
must use\dots This book is written with the intention of bridging that
gap. It was designed to be used in one or more of several possible
ways: (1) as a reference book; (2) as a self-study guide; or (3) as a
textbook for a course in advanced linear algebra, either at the
upper-class undergraduate level or at the first-year graduate level.
The book is self-contained\dots It does, however, assume a seriousness
of purpose and modicum of mathematical sophistication on the part of
the reader.’ The book has twenty chapters with the titles: Notation
and terminology, fields, Vector spaces over a field, Algebras over a
field, Linear dependence and dimension, Linear transformations, The
endomorphism algebra of a vector space, Representation of linear
transformations by matrices, The algebra of square matrices, Systems of
linear equations, Determinants, Eigenvalues and eigenvectors, Krylov
subspaces, The dual space, Inner product spaces, Orthogonality,
Selfadjoint endomorphisms, Unitary and normal endomorphisms,
Moore-Penrose pseudoinverses, Bilinear transformations and forms. There
are a large number of exercises, both numerical and theoretical. The
book is written in a formal mathematical style, but concepts and
theorems are carefully introduced and explained and there are many
examples in the text. Furthermore, the author takes the trouble to
introduce more general structures whenever possible in order to place
definitions and results in their right context. This makes the book a
bit daunting, but the motivated reader will be well rewarded for his
perseverance by finding a very complete coverage of linear algebra and
also a useful introduction to allied algebraic topics. A novel and
welcome feature is the inclusion of nearly a hundred small pictures of
mathematicians connected in some way with linear algebra, along with a
few lines introducing them and explaining the connection. These appear
every few pages at the bottom of a page and shed an interesting
historical light on facets of the subject. This book is a very useful
addition to the relatively small and select set of serious mathematical
introductions to linear algebra and can be warmly recommended to any
graduate student with a more than passing interest in linear algebra.
rv: Rabe von Randow (Bonn)