Lectures on Linear Algebra (Dover Books on Mathematics)

Published just a few months later than Paul Halmos'  Finite-Dimensional Vector Space  (1947), Gelfand's "Lectures on Linear Algebra" (1948), of which the English translation of the revised second Russian edition is the book under review, was the second linear algebra textbook in history.This book was based on an undergraduate algebra course that Gelfand taught in Moscow State University in the early 1940s. When it was written, linear algebra had yet to become an independent subject in universities worldwide. Student in those days usually learned the notion of determinant and how to solve systems of linear equations in a first course on algebra, and were exposed to other topics in a subsequent "higher algebra" course.The content of this book is therefore a bit more advanced than that of a typical modern introductory text. It completely leaves out discussions on systems of linear equations and determinant, but covers many topics that are nowadays usually removed from a first course, including diagonalization of symmetric bilinear form and Hermitian form, Gram determinant, Sylvester's law of inertia, polar decomposition, Courant-Fischer minimax principle, polynomials of Jordan form, elementary divisors and Smith normal form. A brief discussion of tensors is also included in the last chapter.Given its thin size, its rich content is truly amazing. However, its omissions of topics shall not be overlooked. Besides systems of linear equations and determinant, several important topics or notions such as direct sum, quotient space and minimal polynomial are also omitted. Since the book has no subject index and many conventionally named theorems or notions (such as Sylvester's law of inertia and polar decomposition) are unnamed in the book, it is hard for a casual reader to tell what are covered and what are not.As such, I do not recommend any novice learners to use this book as the main text. However, I recommend anyone who wants to consolidate their knowledge of linear algebra to read this book.This book has two two features that I believe are still unsurpassed by other linear algebra texts. The first is the provision of multiple proofs for some theorems. Two proofs are given (pp.137-142, pp.149-163) for the existence of Jordan form over the complex field. Two proofs are given (pp.110-111) for the fact that normal matrices are unitarily diagonalizable. An additional proof (pp.127-129) that a real quadratic form can be orthogonally diagonalized WITHOUT using complex numbers is also provided. Three proofs are given (pp.42-52, pp.67-68) for the fact that a quadratic form can be written as a sum of squares, one by completing squares, one by the method of Jacobi and also one by geometric (orthogonality) argument. All these alternative proofs provide the readers great opportunities to understand the subject from different perspectives.The second feature is the abundance of enlightening explanations. Here are a few examples: * (p.34) "Any 'geometric' assertions pertaining to two or three vectors is true if it is true in elementary geometry of three-space. Indeed, the vectors in question span a subspace of dimension at most three. This subspace is isomorphic to ordinary three space (or a subspace of it), and it therefore suffices to verify the assertion in the latter space. In particular the Schwarz inequality -- a geometric theorem about a pair of vectors -- is true because it is true in elementary geometry." [Read carefully. Gelfand here points out that it suffices to establish an isomorphism between the Euclidean space and a SUBSPACE of an inner product space. Thus the mere fact that Schwarz inequality is true in the Euclidean space -- a finite-dimensional inner product space -- implies that the inequality is true in any INFINITE-dimensional inner product space.] * If A is positive definite and B is Hermitian, they can be simultaneously diagonalized by *-congruence. In other textbooks, this is usually proved in matrix language as follows: let A^{-1/2}BA^{-1/2} be unitarily diagonalized as QDQ^*. Then A=PP^* and B=PDP^* where P=A^{1/2}Q. While this proof is not difficult, the formation of the product A^{-1/2}BA^{-1/2} is unnatural. Gelfand, however, explains in his book (p.41) that an inner product IS precisely a positive definite quadratic form. The aforementioned statement can thus be proved (pp.101-102) in a natural, coordinate-free and succinct manner: just pick an orthonormal basis with respect to the inner product A such that B is diagonalized. * Cayley-Hamilton theorem is proved on pp.88-90 using factor theorem. The proof per se is not new and can be found in many other textbooks, but unlike many authors, Gelfand does care to explain why the factor theorem also needs to be proved (because matrix multiplication is not commutative). * (p.122) "What motivates the division of orthogonal transformations into proper and improper transformations is the fact that any orthogonal transformation which can be obtained by continuous deformation from the identity transformation is necessarily proper. " * (p.143) "We... wish to obtain invariants of a transformation from its matrix, i.e. expressions depending on the transformation alone." (p.181) "With any tensor a_i^j of rank two we can associate a sequence of invariants a_α^α, a_α^β a_β^α, ... ."Among all explanations given in this book, my favorite one is about matrix representations of linear transformations and bilinear forms. Matrix is a useful tool in linear algebra, but it conflates the meanings of linear transformation and bilinear transformation. This sometimes causes confusions even to more experienced students. Gelfand's book is the only one I know that carefully addresses the issue. On pp.91-92, Gelfand explains why, in general, the matrix of a bilinear form has little to do with the linear transformation represented by the same matrix. However, given an inner product, we may associate a bilinear form uniquely with a linear transformation. (A remark similar in spirit about the identification of a vector space and its dual can also be found on p.171.)I have only one complaint about this book. Gelfand has given two proofs of the existence of Jordan form. I like both of them. However, in order to understand Jordan form without messing with arbitrary bases, I still think nilpotence and generalized eigenspace should be taught. Yet, no proof along this conventional line of attack is included in this text.

This is not a book for a first time learner of linear algebra. That being said, this book is excellent for gaining a more rigorous understanding of the subject. The presentation is very clear, and I rarely found myself confused by any wording.

I'm an engineering professor and have started suggesting this book to my students, both grad and undergrad. It's not Currant and Hilbert, but then again you won't strain your back carrying it around.

Very good book! It's not for undergraduate students.

The bilinear (quadratic like) form is related to the inner product in this text: two things that are almost entirely left out in many American linear algebra texts. The text is a translation from Russian andis somewhat difficult with the axiomatic development form used.Explanations, examples and problems aren't a real part of this method of presentation, so the resulting text is more like a graduate textthan the second year texts of American Linear algebra.The price is right for real students of algebra.

The professor who recommended this book made the comment that every time you re-read it, you notice something else that you missed the last time you read it. This is absolutely true.I must say, the first time I picked up this book, I did not like it. The notation was not what I was used to, and the book dives right in, assuming a lot of background (matrices, determinants, etc.) but covering material which many people find boring (bases, etc.). However, when you read deeper, there's a lot here. Once you get past the ugly notation, the proofs are extraordinarily clear. And in spite of the books small size, there is a remarkable amount of motivation and discussion.Like the other reviewer said, this is not a book to learn linear algebra from for the first time: this is an advanced book that is useful for graduate students who have already had a linear algebra course and who want to learn more topics, or understand topics on a deeper level.This is an excellent book; the bottom line is that it's so cheap that there's no excuse NOT to buy it.

As a 9th grade student using this book for self-study, I find this book very clear, concise and easy to follow. Gelfand really emphasises the beauty of this subject. Those saying it is a "graduate text" are clearly wrong. While a small amount of previous linear algebra is necessary, one need not have an entire years worth of it before taking this book on. I highly recommend this book for anyone wanting a thorough understanding of the subject without having to slog through pages of worked examples, easy exercises and unneeded repetition.

I can only strongly recommend this somewhat forgotten textbook to anyone who needs a basic but otherwise thorough exposition to linear algebra. It conflates rigor, examples, and exercises very nicely, although in a very economic manner. Its coordinate-oriented presentation may not appeal to the mathematician in search of ultimate generality, but it is ideal for the physics and engineering-minded person. It is unfortunate that nowadays so many S&E students become first aquainted with rings of polynomials than with rotation matrices! Do not get me wrong, though: the book contains real math inside, but in the best Russian textbook presentation tradition.

Apparently a classic on linear algebra. The book explains concisely vectors spaces with scalar product and linear transformations. The book really provides insight in questions like characterising invariants of linear transformations in the chapter treating the Jordan decomposition. The final chapter on dual spaces is also really clear.