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I'm starting maths degree this September and hoping to do some reading on Linear Algebra before I start. What are some good introductory books?

Is there something that starts from the very beginning but also covers stuff in depth? (Is there a book that will be useful for both 1st year and 2nd year linear algebra modules?)

It would also be great if the book has lots of exercises (proof questions!) and solutions given as well.

Thanks in advance.

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Look at the two books:

  1. Basic Linear Algebra by Blyth and Robertson
  2. further Linear Algebra by Blyth and Robertson.

the second is the sequel to the first. These two books will teach you most of the stuff you will need to know. These books have a lot of exercises with solutions.

Also take a look at the classic by Hoffmann Kunze. It has lot of exercises without solutions.

The book by Axler "Linear Algebra Done Right" is decent but I will not suggest it if you are going to continue your studies towards modules as well. This book banishes determinants and polynomials in the study of linear operators. While this may be helpful over the complex field, this approach has limitations when you go to more general rings and modules.

There is another well written book "Finite Dimensional Vector spaces" by Halmos. I do not remember if it has sufficient exercises though.

For a little more advanced level, take a look at "Advanced Linear Algebra" by Steven Roman and "Matrix Analysis Vol. 1" by Horn and Johnson.

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    $\begingroup$ The Blyth&Robertson texts look nice! Thank you for the recommendation. As well, I think that Halmos has sufficient, very interesting exercises; particularly so when paired with his linear algebra problem book. $\endgroup$ – user59083 May 28 '13 at 5:15
  • $\begingroup$ Thank you. Could you possibly compare: Blyth&Robertson, Axler, Lang in terms of difficulty? $\endgroup$ – mathskid05 Jun 1 '13 at 11:06
  • $\begingroup$ And do you know anything about Friedberg's book? (done some research on this forum and it came up a lot!) Thanks $\endgroup$ – mathskid05 Jun 1 '13 at 11:07
  • $\begingroup$ I do not know about either Lang or Friedberg. However, absolutely love Blyth and Robertson and is very lucid and easy to follow. If you master those two books, you will have a very solid grasp of Linear Algebra. $\endgroup$ – Vishal Gupta Jun 1 '13 at 15:34
  • $\begingroup$ Thanks :) I think I'll go with Blyth and Robertson. $\endgroup$ – mathskid05 Jun 1 '13 at 22:43
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I really liked Linear Algebra Done Right by Sheldon Axler. It starts from the very beginning and contains mostly proof questions; but unfortunately there are no solutions.

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    $\begingroup$ There is a solutions manual floating about on the web. $\endgroup$ – user59083 May 28 '13 at 5:02
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A book that is absolutely terrific for self-study is Halmos's Linear Algebra Problem Book. It's a bit different from most other introductory linear algebra books and it's definitely worth a look. Basically it is about 300 pages of carefully crafted problems with hints and complete solutions. All theory is presented as problems and is then further explained in the solutions. I think it's a very nice approach.

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Lang's Linear Algebra starts at the very beginning and covers stuff in depth.

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  • $\begingroup$ I think Lang could have more exercises/problems. $\endgroup$ – user59083 May 28 '13 at 5:15
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Just an empiric, maybe a little bit personal, paranoic, exagerrated (and not so serious) piece of advice. Take any Linear Algebra book and look at the definition the author gives of a matrix. If it is something that sounds like "A $m\times n$ matrix (over a field) is an array of scalars (numbers) with $m$ rows and $n$ columns" and so on, try to look at another book. If after a while you still can't find anything which gives a different definition for a matrix (and I doubt you will so easily), then pick one of the textbook suggested so far (actually I haven't checked if some of them provides another definition for a matrix). But if you find one which seems to disagree with the fact that a matrix has to be defined as an array of numbers, it might be worth to try and read it.

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  • $\begingroup$ What is your preferred definition of matrix? The one you don't like seems perfectly sensible to me. $\endgroup$ – Potato May 28 '13 at 4:29
  • $\begingroup$ Do you want to define it as a function from $\{1,...,m\}\times \{1,...,n\}$ to $\mathbb{C}$ or whatever space you want? $\endgroup$ – Vishal Gupta May 28 '13 at 4:45
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    $\begingroup$ @Vishal Yes, a $m\times n$ matrix over a field $F$ (or over a ring, if you want more generality) is a function $\bar{m}\times {\bar n}\longrightarrow F$, i.e an element of $F^{\bar{m}\times \bar{n}}$, where by $\bar{m}$ I mean the set $\{1,2,\dots, m\}$. Working with fields, you get immediately that the set of all $m\times n$ matrices forms a vector space under pointwise addition and scalar multiplication and so you don't need to define the sum of two matrices or the product between a scalar and a matrix: you have them for free. $\endgroup$ – Marco Vergura May 28 '13 at 8:14
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    $\begingroup$ @JamesS.Cook This is sick, IMHO. At least if we talk about introductory textbooks like those in Linear Algebra or Calculus which should be basically adressed to first year undergraduate student. Such books should help novice mathematician in developing a critic sensibility to mathematical definitions (one professor of mine once wrote that "the job of a Mathematician is that of defining and demonstrating"), which I do think is essential. There aren't formal and informal ways of doing things: there is just a right and mathematical way or a wrong and non-mathematical one. $\endgroup$ – Marco Vergura Jun 7 '13 at 22:14
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    $\begingroup$ Luckily I took a course in Linear Algebra whose instructor was very careful in stressing importance of rigour and, so to say, formalism. When you discover that $(a,b):=\{\{a\},\ \{a,b\}\}$, you see at once the idea, the intuition behind such a definition and, somehow, its aesthetic. $\endgroup$ – Marco Vergura Jun 7 '13 at 22:22
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I would recommend, arranged somewhat in order of difficulty/sophistication, that you take a look at:

1) Strang Linear Algebra and its Applications (don't forget the MIT OCW lectures)

2) Paul Halmos Finite Dimensional Vector Spaces (the recommendation for his problem book is also good)

3) Hoffman & Kunze Linear Algebra

4) Shafarevich Linear Algebra and Geometry

5) Roman Advanced Linear Algebra

A nice, cheap alternative to Halmos and Hoffman would be Shilov's text published by Dover. There are also nice books that have linear algebra integrated with other material such as Apostol's Calculus (which I highly recommend), Hubbard's vector calculus text or Artin's abstract algebra text.

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