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could someone recommend me pure math linear algebra book?

chickenwin

New member
Feb 13, 2014
12
this probably is a wrong section but i did not know where else to put it (feel free to move it to where it belongs).

that said could someone recommend me a pure math linear algebra book? preferably with LOTS and LOTS of examples! i think i have decent grasp of the basic theory but i have been having much difficulty with some new topics like minimal polynomial, frobenius canonical form, etc and i am wondering if there is a solid book i could be looking at.

cheers.
 

ThePerfectHacker

Well-known member
Jan 26, 2012
236
this probably is a wrong section but i did not know where else to put it (feel free to move it to where it belongs).

that said could someone recommend me a pure math linear algebra book? preferably with LOTS and LOTS of examples! i think i have decent grasp of the basic theory but i have been having much difficulty with some new topics like minimal polynomial, frobenius canonical form, etc and i am wondering if there is a solid book i could be looking at.

cheers.
Most undergraduate on linear algebra give you completely the wrong impression of what linear algebra is about. If your intention is to understand the theory of linear algebra and write proofs with them then I would recommend this. This book avoids the use of determinants and matrices until the very end of the book. It focuses on the mathematics primarily and is intended for people of a mathematics background.

If your intention is to merely learn all the computations in linear algebra then this is not the book for your needs.
 

Guest

Active member
Jan 4, 2014
199
If your intention is to understand the theory of linear algebra and write proofs with them then I would recommend this.
Do you have a similar one for abstract algebra? Thanks.

[Sorry to chickenwin for gatecrashing his question, btw].
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
that said could someone recommend me a pure math linear algebra book? preferably with LOTS and LOTS of examples! i think i have decent grasp of the basic theory but i have been having much difficulty with some new topics like minimal polynomial, frobenius canonical form, etc and i am wondering if there is a soly d book i could be looking at.
There is no correct answer here. I don't agree with sentiments in reply #2.
Here is my recommendation, a text by Larry Smith .
 

ThePerfectHacker

Well-known member
Jan 26, 2012
236
Do you have a similar one for abstract algebra? Thanks.

[Sorry to chickenwin for gatecrashing his question, btw].

A very very gentle start towards algebra is either Fraleigh or Gallian. The disadvantage with these books is that they are very basic on algebra. So even if you read the entire book there will still be a lot of algebra missing. I like this book more, it is a bit more advanced than the other ones, later on in the book the proofs gets more concise and they assume you can follow them.

A very comprehensive book on algebra that has a lot of stuff in it and contains much algebra is by Dummit and Foote. This is a great book that is written like a textbook and has a ton of algebra in it, but it is tougher than any of the other ones I posted. It can even be used for many graduate courses.

I guess I should mention an awful algebra book but it is huge. I am talking about the book by Lang. As a textbook it is awful and I never read anything by Lang that I liked. Non-motivational, dense, boring. Really intended for mathematicians who already know algebra.
 

chickenwin

New member
Feb 13, 2014
12
Do you have a similar one for abstract algebra? Thanks.

[Sorry to chickenwin for gatecrashing his question, btw].
thanks for the recommendation guys. keep them coming (heart)

@ guest, i have this and i like it quite a bit
A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics): Charles C Pinter: 9780486474175: Amazon.com: Books

also i recall this being pretty decent when i took the course
Contemporary Abstract Algebra: Joseph Gallian: 9780547165097: Amazon.com: Books
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
There is no correct answer here. I don't agree with sentiments in reply #2.
Here is my recommendation, a text by Larry Smith .
Hi Plato.

Larry Smith's book looks like a possible good text ... but I note it has a couple of bad reviews on Amazon (the hardcover 1998 book, anyway) - presumably you feel these reviews are misguided?

Peter
 

Guest

Active member
Jan 4, 2014
199
A very very gentle start towards algebra is either Fraleigh or Gallian. The disadvantage with these books is that they are very basic on algebra. So even if you read the entire book there will still be a lot of algebra missing. I like this book more, it is a bit more advanced than the other ones, later on in the book the proofs gets more concise and they assume you can follow them.

A very comprehensive book on algebra that has a lot of stuff in it and contains much algebra is by Dummit and Foote. This is a great book that is written like a textbook and has a ton of algebra in it, but it is tougher than any of the other ones I posted. It can even be used for many graduate courses.

I guess I should mention an awful algebra book but it is huge. I am talking about the book by Lang. As a textbook it is awful and I never read anything by Lang that I liked. Non-motivational, dense, boring. Really intended for mathematicians who already know algebra.
Thanks for the reply and recommendations. I appreciate it. I've heard good things about a book that you haven't mentioned: Herstein's Topics In Algebra. Are you familiar with it, and if so, any comments?
 

chickenwin

New member
Feb 13, 2014
12
Hi Plato.

Larry Smith's book looks like a possible good text ... but I note it has a couple of bad reviews on Amazon (the hardcover 1998 book, anyway) - presumably you feel these reviews are misguided?

Peter

i'm curious about this as well. bad reviews are the first thing i noticed.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Thanks for the reply and recommendations. I appreciate it. I've heard good things about a book that you haven't mentioned: Herstein's Topics In Algebra. Are you familiar with it, and if so, any comments?
Ah, my little blue book. It has long been a favorite of mine.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,193
Thanks for the reply and recommendations. I appreciate it. I've heard good things about a book that you haven't mentioned: Herstein's Topics In Algebra. Are you familiar with it, and if so, any comments?
I'd actually recommend Herstein's Abstract Algebra for the beginner. I enjoyed it immensely, and found it quite readable. It's a great prep for Dummit and Foote, because both books use extremely similar notation.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Addressing the original question:

Axler's book is oft-recommended. Two others worth investigating:

Introduction to Linear Algebra, Gilbert Strang
Linear Algebra (2nd ed) Hoffman & Kunze

Some general ramblings:

1. Vectors aren't as interesting (and more predictable than) as linear transformations.

2. Some bases are easier to work in than others. Linear transformations can help us pick "the right basis".

3. Use a basis only as a last resort: a choice of basis is often "too particular"; if something is true, it ought to be true "for any basis", and if we can find a "basis-free" way to prove something, such a proof is more powerful (and often more illuminating).

4. Don't be afraid of quotient spaces. These are just "collapsed" vector spaces which are the image of some singular linear transformation. The null space of the linear transformation is a measure of "how severe" the collapse is.

5. When in doubt, try to find the dimensions. Often this reduces a problem to simple arithmetic of integers.

6. Use linearity often, it is what linear algebra is ALL ABOUT. Linearity is "nice", many problems in math have solutions that go:

a). First, take the linear case:......

b) Next, suppose that there is a linear approximation.....

(after which you use part (a)).
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Ah, my little blue book. It has long been a favorite of mine.
:)
One of my favourite books too (of all the books I have ever read be it Mathematics or non-Mathematics books). For me it's a little red book.