Linear Algebra

by Serge Lang

Linear Algebra is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorems for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finie-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants, and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.

Genres: Mathematics, Textbooks, Algebra, Nonfiction

296 pages, Hardcover

First published January 1, 1970

About the author

Serge Lang was an influential mathematician in the field of number theory. Algebra is his most famous book.

Librarian Note: There is more than one author in the GoodReads database with this name. See this thread for more information.

Reviews

[Z. N. · Rating 5 out of 5]

A fantastic little book with some very slick proofs. Worth reading if only for the very eye-opening alternate proofs presented (or at least outlined) alongside the classical ones, such as a continuity argument to prove the Cayley-Hamilton theorem in the non-diagonalisable case, and Lagrange multipliers (of all things) to prove the existence of an eigenvector for a symmetric matrix. I'm surprised I've never thought about applying Bezout's Lemma (from number theory) to polynomials, and even more surprised to see the number of linear algebra results it can prove.

The chapters flow well, though some parts feel a little rough on the edges; there are some changes in tone, and some minor typos - but they don't really detract from the text.

From a modern standpoint, it was a little jarring to read Lang claiming that determinants were "efficient" - this might be the only place where one is lead astray by letting n be 2 or 3.

Readers wanting more than a brief overview of modern applications (Lang treats homogeneous linear ODEs as a single example of primary decomposition, and recurrences aren't mentioned at all) will want to consult some more modern sources.

[meadow · Rating 3 out of 5]

perhaps it was the class i used this book for, but i thought this book wasn't the most well organized. i don't know, something about it struck me as not cohesive, but the more i think about it the more i think it's just the fault of the instructor i had for the course, since we didn't even use the book very much. and maybe it's also my natural aversion to linear algebra. in any case i don't think it's one of lang's best books. good exercises though. as i recall it's sometimes lacking in examples in a few spots where you really want them, but for the most part is good about examples.

[Giuseppe Sguera · Rating 5 out of 5]

Letteralmente la Bibbia.
Il miglior testo per chi voglia studiare e approfondire l'algebra lineare.
Una trattazione completa ed esauriente della teoria degli spazi vettoriali.
Il testo fornisce anche le basi della teoria dei gruppi e degli anelli per coloro che intendono proseguire lo studio dell'algebra.
L'esposizione è chiara in tutte le parti.
Consigliatissimo per studenti di matematica, fisica e (eliminando dalla trattazione alcuni argomenti troppo astratti) persino per i corsi di ingegneria

[Jose Ignacio · Rating 4 out of 5]

Well, it seems that this books is for mathematicians and I'm not one, so it seems I would prefer one oriented to applications. Anyways, a good read.

[Coda Di Lupo · Rating 5 out of 5]

read

[Anil · Rating 5 out of 5]

The book is excellent. especially worth is the appendix odds and ends which delas with angles great!