Michael Artin's Algebra

[Howers]

What is the necessary background to take on Michael Artin's Algebra book?

And is it harder than Spivak's Manifolds?



And + and why is linear algebra taught before abstract algebra. Isn't it more logical the other way around?

 

[HallsofIvy]

Mostly what you need is "mathematical maturity"- that is, the recognition of the abstraction of mathematics and at least some facility in understanding and writing proofs.

I would strongly recommend taking Linear Algebra before Abstract Algebra. It gives an introduction to the methods and concepts used in Abstract Algebra with very concrete examples.

 

[Howers]

If it's mathematical maturity, I believe I have that - atleast in the analysis sector. My linear algebra is not honors level though. Would that be enough, or should I master LA beforehand. I didn't think abstract relied on linear, except when discussing examples.

 

[d_leet]

Artin's book covers a great deal of what is usually taught in a linear algebra course, as well as the first chapter covers the basics you need to know about matrices etc. so I think as long as you are comfortable with proofs and have the so called quality of "mathematical maturity" then you should be fine to read Artin's book.

 

[mathwonk]

mike wrote and taught his book for sophomores at mit.

it may be true that elementary linear algebra is best taught before abstract algebra, but jordan and rational canonical forms of matrices are very hard to grasp unless you have had some exposure to cyclic and abelian groups first.

i.e. the classification of finite abelian groups as products of cyclic groups is a very good prerecquisite to understanding advanced linear algebra.

so maybe the right order would be elementary linear algebra, then elementary abstract algebra, then advanced linear algebra, then advanced abstract algebra. in fact we teach it this way at georgia.

 

[n_bourbaki]

The teaching of linear algebra before abstract algebra in US universities is probably a choice born of pragmatism, and necessity for non-mathematicians. After all, how many people in those math classes are actually mathematicians, and how many are engineers?

Coming through a different system, for example, I met Sylow's theorems before Jordan canonical form.

 

[mathwonk]

if you think about it nb, that makes no sense. sylow's theorem is about the more difficult non abelian group actions, while jordan form is a consequence of the easier abelian group theory which follows from the euclidean algorithm.

it is just a tradition, promulgated by commonly used books like herstein and hungerford, that objects like groups with one operation are taught before objects with two operations, like rings fields and linear algebra. but it is questionable pedagogy, whether for math majors or non.

 

[n_bourbaki]

But how can we explain that a field is an abelian group under addition, and where the non-zero elements are an abelian group under multiplication if we don't discuss groups first?

And an introduction to linear algebra came before that. Just restricted to R^2 and R^3.

 

[mathwonk]

1) of course it does make sense to define a group first, but it does not follow it makes sense to prove sylows theorems.

2) that is a good way to do things, linear algebra in R^2 and R^3 first.

 

[uman]

The author explicitly states that calculus is a prerequisite, as well as a familiarity with "the basic properties of complex numbers".

 

[Howers]

So I gather linear algebra is not needed?

If one has a basic grounding in vector spaces, linear operators, basis and change of basis, matrix representations, bilinear forms, quadratic forms, symmetric bilinear forms, Hermitian forms, different ways to diagonlize (orthogonal, idempotent), general projections, inner products, and introductory canonical forms Artin can be done?

 

[morphism]

So I gather linear algebra is not needed?

If one has a basic grounding in vector spaces, linear operators, basis and change of basis, matrix representations, bilinear forms, quadratic forms, symmetric bilinear forms, Hermitian forms, different ways to diagonlize (orthogonal, idempotent), general projections, inner products, and introductory canonical forms Artin can be done?

What is linear algebra if not that?

 

[Howers]

What is linear algebra if not that?

I already said I had non honors linear algebra. I meant is advanced linear algebra needed? Like something out of Insel/Friedberg.

I heard Artin's book leaves everything to the reader so I don't know if a basic survey of linear algebra is enough. All I know is a lot of people either love it or hate it, and nothing in between. More people hate it I think, so I'm thinking it needs a strong background.

 

[morphism]

I think the most reasonable thing you can do is get a copy of the book (from e.g. your school's library) and judge for yourself. Even if you have the necessary background, that doesn't mean it will be a good read. Fortunately there are many good algebra books out there that can serve as alternatives. For instance, try to see if you can find Herstein's Topics in Algebra -- it should be worth a look as well.

 

[mathwonk]

artins book certainly does not assume any knowledge of linear or any other college level algebra. and it bends way over backward to explain things in a very elementary and clear way. still some things are given to the reader to do, like read it, think about it, and try to understand it.

there is no better book written for strong, naive students.

it is very different in tone from insel and friedberg, which is not a prerecquisite.

on the other hand if you like insel and friedberg a lot, you may not like artin, at least not yet.

insel and friedberg is a book for "dummies", that cuts down all the trees and lays them out for you in a long line, totally obscuring the forest. artin on the other hand teaches the essential ideas rather than the details.

but if insel and friedberg seems great to you, then go ahead and use it, and then try artin again and see if the added knowledge from insel et al, helps you get up on the non dummy level. we all start out as dummies, but it is recommended to try not to stay there.

 

[Howers]

insel and friedberg is a book for "dummies", that cuts down all the trees and lays them out for you in a long line, totally obscuring the forest. artin on the other hand teaches the essential ideas rather than the details.

but if insel and friedberg seems great to you, then go ahead and use it, and then try artin again and see if the added knowledge from insel et al, helps you get up on the non dummy level. we all start out as dummies, but it is recommended to try not to stay there.

heh, not sure what you are saying about insel being for dummies. are you saying insel obscures the main point by providing too much detail? or that it is at a low level?

i would say i prefer a detailed treatment over a skimping of the essentials. i like the axiomatic approach to math. so is artin or dummit good for that?