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Finite-Dimensional Vector Spaces
From the "The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity....The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher." --ZENTRALBLATT FÜR MATHEMATIK
210 pages, Hardcover
First published January 1, 1947
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Displaying 1 - 11 of 11 reviews
September 1, 2021
I own at least a dozen books on various aspects of linear algebra, ranging from community college level textbooks (Anton) to treatments targeting specific industries such as computer graphics. Invariably, I end up poring over the relevant material in Halmos to ensure that both the exposition in some other text and my understanding of the topic is correct.
Time spent reading this relatively thin book is measured in "hours per page." It's time well spent.
July 25, 2020
Linear algebra occupies an ambiguous place in the curriculum. Though it belongs at the same level as introductory calculus, it is not normally taught in high school. This reviewer took it in the fall semester of his freshman year at Princeton, as part of the coursework designed for honors students. The textbook for the course was Paul Halmos’ Finite-Dimensional Vector Spaces, in the Springer series of undergraduate texts in mathematics. The reviewer has fond memories of that course taught by then-recent Fields medalist Gerd Faltings as, more than any other, it made him into a mathematician. In retrospect, it was not apparent to him then, as it is now, how distinctive Halmos’ approach to his subject is. Most treatments of linear algebra at the beginning level are workmanlike but largely computational in emphasis; Halmos is a genuine pure mathematician writing for mathematicians, or for those who would become such. As such, his style is adapted to their needs.
The difference is style is apparent at the outset. Halmos does not jump immediately into Cartesian, or Euclidean coordinates. Rather, he builds up the subject in a coordinate-free manner from scratch starting with the axioms defining a field and a vector space over a field. The notions of linear independence, basis and subspace follow. For this reviewer, it struck him as nothing less than an epiphany, sitting in Fine library during the first week of classes, to encounter the abstract definition of linear independence of a collection of vectors. He saw, perhaps for the first time, where a logical implication was needed to establish something interesting to know (the collection of vectors is linearly independent if for any linear combination to equal zero implies that the coefficients vanish identically). Besides this, it was pleasing and highly revelatory to see how an algebraic formalism could be used to get control over a geometrical property, viz., the dimension of the space spanned by the collection of vectors. From here on, he could tell that Halmos would offer an enjoyable ride. Before even getting to matrices or linear transformations, Halmos continues with linear functionals, annihilators, dual spaces, reflexivity, direct sums, quotient spaces, tensor products and bilinear, multilinear and alternating forms.
In the second chapter, Halmos arrives at last at linear transformations of a vector space, defined in the first place abstractly before introducing matrices. The treatment focuses on concepts needed to understand linear transformations as operators, such as invariant subspaces, reducibility, similarity, projections, nilpotence, adjoints, range and null spaces, proper values and multiplicity. The chapter culminates in a very instructive proof of the Jordan normal form. The distinctiveness of Halmos’ approach comes to the fore here; there is little in the way of the computations one might expect from an introduction to linear algebra, e.g., how to find determinants, inverses, the solution of an inhomogeneous linear system of equations, eigenvalues and eigenvectors etc., although of course a number of the homework exercises call upon the student to go into these matters. Rather, Halmos wants to lead the reader to think like a pure mathematician interested in characterizing the behavior of linear transformations of vector spaces, in general.
The third chapter concerns what happens once one introduces an inner product on a vector space and the associated concept of orthogonality. One arrives at the Cauchy-Schwartz inequality and the idea of a complete orthonormal set. The heart of the chapter is an efficient two-page proof of the spectral theorem for normal transformations. The fourth and last chapter goes a little bit into analysis, concerning the norm and convergence of sequences of vectors and linear transformations. These ideas receive application to the minimax principle, the ergodic theorem of von Neumann and power series of linear transformations. A pedagogically helpful appendix describes what happens when one generalizes finite-dimensional vector spaces to infinitely many dimensions (Hilbert space) and some of the complications one encounters in the more general setting.
As will be apparent from this listing of the principal contents of this little textbook, the perspective is everywhere that of the pure mathematician who has in his mind’s eye the eventual application of the ideas introduced here to functional analysis and operator theory. Hence, there is comparatively little stress on computational tasks such as loom large in many beginning textbooks on linear algebra. The care with which Halmos selects and arranges his material will perhaps be lost on the first-time reader, but leaps out at one if one goes back to it after having had exposure to the graduate-level curriculum. All told, the author accomplishes his objective to demonstrate the simple and elegant geometric notions that emerge when one applies to the finite-dimensional case the powerful techniques originally worked out to handle problems in infinitely many dimensions, reaching back to Hilbert himself and the theory of integral equations around the turn of the twentieth century. One can well understand why this work, which might be considered an idiosyncratic choice for a first encounter with linear algebra, was selected for honors mathematics majors at Princeton.
All through the text, Halmos’ engaging prose style stands out. There are well over three hundred exercises included. Many of these might be regarded as being on the easy side, at least for an advanced student, but there are a good number of rather hard problems as well to challenge such a one. This reviewer remembers spending many hours, even days, to puzzle out the solution to the hardest of the exercises.
The difference is style is apparent at the outset. Halmos does not jump immediately into Cartesian, or Euclidean coordinates. Rather, he builds up the subject in a coordinate-free manner from scratch starting with the axioms defining a field and a vector space over a field. The notions of linear independence, basis and subspace follow. For this reviewer, it struck him as nothing less than an epiphany, sitting in Fine library during the first week of classes, to encounter the abstract definition of linear independence of a collection of vectors. He saw, perhaps for the first time, where a logical implication was needed to establish something interesting to know (the collection of vectors is linearly independent if for any linear combination to equal zero implies that the coefficients vanish identically). Besides this, it was pleasing and highly revelatory to see how an algebraic formalism could be used to get control over a geometrical property, viz., the dimension of the space spanned by the collection of vectors. From here on, he could tell that Halmos would offer an enjoyable ride. Before even getting to matrices or linear transformations, Halmos continues with linear functionals, annihilators, dual spaces, reflexivity, direct sums, quotient spaces, tensor products and bilinear, multilinear and alternating forms.
In the second chapter, Halmos arrives at last at linear transformations of a vector space, defined in the first place abstractly before introducing matrices. The treatment focuses on concepts needed to understand linear transformations as operators, such as invariant subspaces, reducibility, similarity, projections, nilpotence, adjoints, range and null spaces, proper values and multiplicity. The chapter culminates in a very instructive proof of the Jordan normal form. The distinctiveness of Halmos’ approach comes to the fore here; there is little in the way of the computations one might expect from an introduction to linear algebra, e.g., how to find determinants, inverses, the solution of an inhomogeneous linear system of equations, eigenvalues and eigenvectors etc., although of course a number of the homework exercises call upon the student to go into these matters. Rather, Halmos wants to lead the reader to think like a pure mathematician interested in characterizing the behavior of linear transformations of vector spaces, in general.
The third chapter concerns what happens once one introduces an inner product on a vector space and the associated concept of orthogonality. One arrives at the Cauchy-Schwartz inequality and the idea of a complete orthonormal set. The heart of the chapter is an efficient two-page proof of the spectral theorem for normal transformations. The fourth and last chapter goes a little bit into analysis, concerning the norm and convergence of sequences of vectors and linear transformations. These ideas receive application to the minimax principle, the ergodic theorem of von Neumann and power series of linear transformations. A pedagogically helpful appendix describes what happens when one generalizes finite-dimensional vector spaces to infinitely many dimensions (Hilbert space) and some of the complications one encounters in the more general setting.
As will be apparent from this listing of the principal contents of this little textbook, the perspective is everywhere that of the pure mathematician who has in his mind’s eye the eventual application of the ideas introduced here to functional analysis and operator theory. Hence, there is comparatively little stress on computational tasks such as loom large in many beginning textbooks on linear algebra. The care with which Halmos selects and arranges his material will perhaps be lost on the first-time reader, but leaps out at one if one goes back to it after having had exposure to the graduate-level curriculum. All told, the author accomplishes his objective to demonstrate the simple and elegant geometric notions that emerge when one applies to the finite-dimensional case the powerful techniques originally worked out to handle problems in infinitely many dimensions, reaching back to Hilbert himself and the theory of integral equations around the turn of the twentieth century. One can well understand why this work, which might be considered an idiosyncratic choice for a first encounter with linear algebra, was selected for honors mathematics majors at Princeton.
All through the text, Halmos’ engaging prose style stands out. There are well over three hundred exercises included. Many of these might be regarded as being on the easy side, at least for an advanced student, but there are a good number of rather hard problems as well to challenge such a one. This reviewer remembers spending many hours, even days, to puzzle out the solution to the hardest of the exercises.
September 5, 2024
A pleasure to teach from. Concise and engaging.
February 8, 2020
This is how linear algebra should be taught
March 31, 2020
Clear, intuitive, beautifully presented. I loved this book. Truly made me appreciate aspects of linear algebra I thought it too late to enjoy after plowing through one too many dull matrix proofs.
Want to read
August 23, 2020Used for refreshing linear algebra once in a while.
May 15, 2024
This book changed my perspective on what linear algebra is. It is a bit hard at time, so I recommend supplementing this with the Linear Algebra Problem Book by Halmos
May 18, 2021
I would have given the book five stars if I understood it more.
I found that Eliot Nicholson's YouTube channel's playlist on Linear Algebra was much more understandable for me. There are 48 videos, and it is really efficient if you watch them in 2x playback speed.
I found that Eliot Nicholson's YouTube channel's playlist on Linear Algebra was much more understandable for me. There are 48 videos, and it is really efficient if you watch them in 2x playback speed.
August 22, 2018
Like Halmo's book "Linear Algebra Problem Book", this provides the reader with truly amazing insights!
You simply can't go wrong with it, although it must be said that more modern treatments of linear algebra could be more appropriate. Even if the reader prefers more recent books, I would highly recommend to use this as a supplement.
You simply can't go wrong with it, although it must be said that more modern treatments of linear algebra could be more appropriate. Even if the reader prefers more recent books, I would highly recommend to use this as a supplement.
August 14, 2011
Tres bon livre mais il manque des infos sur les polynomes d'endomorphismes, ce qui est dommage.
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March 21, 2014512.523 H194 1974
Displaying 1 - 11 of 11 reviews