Linear and Abstract Algebra textbooks

Icebreaker

Main Question or Discussion Point

I need some recommendations for a good linear algebra textbook, something that's actually used in schools. I've finished linear 1 and 2 and I'm doing some preparation during the summer.

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quasar987
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This might not be what you're looking for but here's one that I'm enjoying at the moment: http://www.math.miami.edu/~ec/book/

I'm really not very advanced in it (pp.31 of 146) but I've browsed a lot and from what I understand, the author used the first half of the book (75 pages) to enunciate the main concepts and results about groups, rings and matrices that he's gonna need for his complete study of linear algebra in other half of the book.

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mathwonk
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Icebreaker
Checking them out as I type this. My thanks.

mathwonk
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i especially recommend sharipov.

thanks mathwonk...i've bene looking for a mathforum

quasar987
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Dang that's a lot of ebooks! Got anymore links mathwonk? What about ebooks on set theory and (introductory) Fourier theory?

Icebreaker
Yes, about set theory... in what class were we supposed to be formally introduced to it? What I've learned about it so far is from a collection of subtle exposures in other courses. The main idea and fundamentals I had to read on my own.

mathwonk
Homework Helper
I learned set theory in high school from a book by allendoerfer and oakley (principles of mathematics), and books by erich kamke and paul halmos (naive set theory).

Since then I have enjoyed the book of Felix Hausdorff, and the original work by Georg Cantor (contributions to the theory of transfinite numbers.).

a nice work on fourier series is the little benjamin (not an e) book by robert t seeley.

i only know the linear algebra ebooks from having to teach the course and needing a reference for my students.

a suitable one is hard to find. i.e. most decent books assume students know how to read and do proofs, but most students today do not.

the ones i listed run the gamut from mickey mouse (matthews) to excellent (sharipov), and i let the reader choose his own poison.

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quasar987
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Hey Icebreaker, what uni do you go to in montreal? I go to Université de Montréal.

Icebreaker
McGill LENTHENING MY MESSAGE TO MEET MINIMUM LENGTH REQUIREMENT

Hurkyl
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Anything wrong with "I go to McGill"?

quasar987
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How many sessions have you completed Ice? And what program are you studying in?

Mathwonk: It is true that we don't know how to read and do proofs. There are many proofs in my analysis textbook that completely eludes me, and most proofs that are not straightfoward, I cannot do on my own. And I know it is the same for most of my classmates. What do you think is different now that makes it so students don't know how to read and do proofs?

mathwonk
Homework Helper
high schools no longer teach euclid. that is the main thing i guess.

also there is a general response that if it is difficult to teach, then just stop trying to teach it, "they'll get it in college".

the official response in the US to the poor high school preparation we provide, is to assume colleges can magically make up the deficit. this is leading to a watering down of what used to be the best higher education system in the world, to one that accomodates the weakest graduates of US high schools.

US college education is now what used to be US high school education. High school is now essentially nothing at all except at a few excellent schools. Students used to learn to read and write and reason logically in high school, now many college bound students do not learn any of these.

High school courses designed to prepare a person to take a certain test, are a joke compared to courses that were intended to teach people to read critically, understand, and write intelligently.

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Icebreaker
quasar987 said:
How many sessions have you completed Ice? And what program are you studying in?
Math, but I switched over from another program. Long story.

Hurkyl said:
Anything wrong with "I go to McGill"?
I suppose there's a reason for the 10 character minimum.

quasar987
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Gold Member

Ice: I have two friends there who just completed their first year. William and Tayeb. Do you know them?

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mathwonk
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it is certainly a challenge. how can we combat the tendency to market education, via appealing only to the greed of people, such as by "hope scholarships" based on falsely inflated grades in high school, as opposed to "merit scholarships" as in the 60's, based on competitive tests?

In the 1960's president kennedy actually inspired people to want to achieve more. remember his speech: " ask not what your country can do for you, but what you can do for your country." where is the leader today who inspires people to achieve and to give, rather than to covet? not to mention m.l.k., who inspired people to risk life and limb for freedom and equality. it is so disturbing that thes people were murdered that i hesitate to draw the obvious conclusions as to who had the most to gain from their disappearance from the political scene.

It is not just sad, it is crucial to the future of our world, for the younger generation to take control of their future, and not let it be stolen from them. In particualr, anyone who wants an A for a course he has not understood is marching to the same tune as a crooked politician trying to divert taxes to his own gain.

I am very encouraged by most of the posts on this forum, for they give evidence of a genuine desire to learn and improve oneself by todays young students, and some not so young. to do ones best, and help others to do theirs, is to fight the good fight. more power to you.

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mathwonk:

Thanks for posting those links to the free books. I took a look at the first few pages of Sharipov's book, and saw a footnote that said the axioms for the the operations of vector addition and scalar multiplication are not independent! I was surprised to see that. Is this common knowledge? I thought maybe Sharipov had come up with his own set of axioms to make things as general as possible, but it turns out Gelfand uses the same ones in his little book from the 60's.

Icebreaker
quasar987 said:
Ice: I have two friends there who just completed their first year. William and Tayeb. Do you know them?
Mmm, unfortunately no.

quasar987
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*tear*

mathwonk for president!

Does anyone know if these titles are good as introductions?:

- "A First Course in Abstract Algebra (6th Edition)" by John B. Fraleigh
- "The fascination of groups" by F. J Budden

They are recommended reading material when taking Algebra here at the local university.
But I'm not sure about these, because the lecturer isn't that good of a teacher. His lecture notes are filled with errors and he also recommended "Algebra" by Serge Lang for an introductory course in Linear Algebra, which seems a bit extreme..

Anyway, feel free to name some other good introductory titles on the subject. I prefer real books to ebooks.

Thanks.

mathwonk
Homework Helper
serge lang's algebra, advanced as it is overall, may actually have some useful discussions about topics in linear algebra.

lang writes extremely well locally at times. I.e. on a single page, devoted to a single idea, he sometimes has the most succint and clear explanation you can find. then the next sentence may make no sense at all.

i would guess your teacher actually meant to recommend one of langs linear algebra books, some of which are extremely clear, and well organized, at least the edition that i saw years ago. of course his books appear repeatedly in many revised forms, and as students get weaker he watered them down more to accomodate.

but lang's book had sections entitled: "the linear map associated to a matrix". and then: "the matrix associated to a linear map".

these are the key constructions, and they are not always that clearly laid out everywhere.

for example people rave about strang and his "4 fundamental subspaces" but i am not so enchanted with them or him myself.

indeed there are really only two subspaces, kernel and image, but there are two maps associated to a matrix, one to the original matrix and one to its transpose, so you get 4 subspaces.

there are two ways to choose a book, either it is has the best discussion by the most expert author, or it describes the material in a way that speaks to you the reader. a learner often needs both kinds, but no one else can recommend the second kind. you must go to the library and compare books.

the purpose of the book that speaks to you, is to unable you eventually to read the one written by the expert.

fraleigh is an ordinary author of ordinary books, whose (expensive) books are liked by some people, lang is an expert author, but one who writes book at many levels.

if you really want a good book, try bourbaki on linear algebra. it is even translated into english, and all bourbaki books include a historical discussion, and a good one, not just the little thumbnail bio with a funny looking picture of newton or leibniz.

bourbaki books are fantastically clear especially in linear algebra and commutative algebra, having been written apparently by the world's best experts in the 60's on algebraic geometry.

for undergraduate linear algebra, i like the book of adams and shifrin, for its mingling of geometry with linear algebra, although the emphasis is actually strongly on matrices. the problems are excellent as in all of shifrin's books.

charles curtis' book was recommended to me by a student and looked pretty good as i recall.

but the ebook of sharipov is really quite well done, with a carefully graduated treatment of the essential idea (nilpotency) underlying the more difficult decomposition theorems, and beginning with some elementary material on sets and functions if i recall correctly.

if you like hardbound books, and have a finite budget, that is another reason to look at lang's book, as it is about 1/2 the price of the more popular ones, like strang and shifrin - adams.

notice (sighh...) that sheldon axler's "linear algebra done right" sells for about 1/3 the price of books on linear algebra done wrong.

there are also some good reprinted paperbacks that sell for around 7 - 10 dollars.

e.g. shilov $15, walter nef$8. how can you go wrong at these prices? and these are good books.

these cheap old reprints are 1960's books which were not written to compensate for the weaknesses of todays students, now their only "shortcoming".

oh, and the absolute classic american linear algebra book, done right, is the one by hoffman and kunze, hands down the best. this is probably the only linear algebra book (except bourbaki), that i would bother to have on my own math book shelf, as opposed to my textbook shelf.

another excellent book, essentially unfindable, is the SMSG paperback on linear algebra written for high schools when the "new math" movement was attempting to recast high school math instruction at a higher level.

this whole effort, although excellently represented by great textbooks, was sidetracked by a lack of funding for teacher preparation, and by a reluctance of the program's architects to infringe on the marketplace by actually selling large numbers of the terrific books they produced.

the idea was to energize the american book marketplace and inspire other authors and publishers to produce good texts. this did not happen, and the "new math" books produced by the american marketplace were execrable, and togetehr with the inadequate instruction from unprepared teachers, helped give the movement a very bad name. the original works from SMSG via Yale, are still outstanding but available only on library shelves of schools of education.

summary: the really good books are by bourbaki and hoffman - kunze.

excellent cheap books are by sharipov, shilov, nef.

good traditional texts are by lang and adams - shifrin, at $50-$100 or more.

there is no way i would pay anywhere near \$100 for fraleigh myself.

a hint: rather than focusing on reducing matrices accurately, learn what the maps are doing. I.e. learn to understand the part your calculator cannot do for you. and learn the proofs.

mathwonk
Homework Helper
shifrin has a nice abstract algebra book, something like "algebra, a geometric approach". and mike artin has a great, if challenging, abstract algebra book, called simply "algebra". we could probably use artin for our neginnig grad course, but he sues it at MIT for sophomores, and I know people who survived it who were really not that strong.

Hm..I must say that I'm only taking the first big steps towards these subjects. And indeed you're absolutely right about these 2 kinds of books.

I think I'm more in need of the second kind of books right now. That's why I bought Strang's book (yes..). By watching his video lectures I've gained more understanding about linear algebra than with any other lectures/book I've seen before. Even if they were in my own language.

Actually, it used to be the case that when studying physics here there was only 1 algebra course available and it covered "vector spaces", "matrices" and a bit of "group theory". These subjects are written over 120 pages of lecture notes and that's that. I heard from students that if you learn these lectures from heart, you pass. But in the end they don't really understand the subject. So I guess that to fill in the (huge) gaps the teacher recommended Lang's "Algebra".
But since you say that Lang is not always so clear..I don't think I'll be able to get a good understanding of the subject with his book, considering my limited foreknowledge. Maybe this book is good as a future project..

Anyway, now they introduced a second algebra course covering things like "set theory", "polynomials", "arithmetics" and again "group theory". That's where the 2 books I listed should come in.

So I think I may have linear algebra covered for now, now I'm in need of a book that corresponds to this new algebra course.

I'll take a look at these books you suggested mathwonk, thanks a lot!

edit: typo

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quasar987