Lectures on Linear Algebra

by Israel M. Gelfand, Abe Shenitzer (Translator)

Prominent Russian mathematician's concise, well-written exposition considers n-dimensional spaces, linear and bilinear forms, linear transformations, canonical form of an arbitrary linear transformation, and an introduction to tensors. While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector.

Genres: Mathematics

208 pages, Paperback

First published September 1, 1989

About the author

Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (Yiddish: ישראל געלפֿאַנד‎, Russian: Израиль Моисеевич Гельфанд ; 2 September [O.S. 20 August] 1913 – 5 October 2009) was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the Wolf Prize, he was a Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University.

Reviews

[Mike Del Vecchio · Rating 5 out of 5]

I enjoy the style and narrative in I.M. Gel'fand's "Lectures on Linear Algebra." The book is from 1948 Russia. You are taken through n-dimensional spaces and into understanding linear transformations (in their canonical form). The final chapters cover dual spaces and tensors.

With L.A., there are different approaches to aid in understanding.
1. A vector-space-granularity driven approach (useful for engineers, but can be grating after a bit)
2. A more theory-based sort of esoteric approach (seems preferable and ideal, but it is difficult to articulate complex concepts with elegance into hard terms with exactitude).

Gel'fand's book is a great balance of both approaches. He gives you everything you might want in his explanations. He goes into some trippy areas with his chapter on complex n-dimensional spaces. If given the choice to only take one Linear Algebra book with me to a tropical island, I may choose this one. There's a lot to consider here.

While other Dover Math books can be hit or miss, in that some are too simple or incomplete, this book is thorough and fulfilling in what it covers. Great for a rainy day. I comfortably sailed through about half of this one.

[Jake · Rating 5 out of 5]

I sat down and read this in a day. I don't understand it yet. I don't have much intuition about what Gelfand speaks of. What I am sure of is that Gelfand saw algebra as a tool for combinatorics and analysis. Many people will see the index notation, as opposed to the matrix notation often used today, and be scared away but there's value in this approach. Especially if you're looking to use algebra for applied work. In fact I'm quite sure at how spartan this text is (under 200 pages) that it is really a tool to get as quickly as possible to combinatorialization.

What this text talks about that I haven't seen much treatment to is bilinear form and intuition one gains on triangularization for combinatorics (he doesn't directly talk about this but it seems implied). As an undergraduate I haven't been able to find a good introduction to tensors until I ran across this book and again, I don't quite understand it yet but the treatment feels right.

[K Sadov · Rating 4 out of 5]

Concise but easy to follow. I wouldn't recommend it as someone's first linear algebra book, but it's a useful refresher for someone who dimly remembers their undergraduate lin alg coursework.