## Linear and Abstract Algebra textbooks

He didn't give it, but I attempted to come up with one myself. Here are the axioms

for vectors u,v,w in a vector space V and numbers a,b in a field K:

1) u + v = v + u
2) (u+v)+w = u+(v+w)
3) There is a zero vector O s.t v + O = v for all v
4) for every v and every O, there is an "opposite vector" v' s.t v + v' = O
5) a(bv) = (ab)v for all a,b, and v
6) (a+b)v = av + bv for all a,b, and v
7) a(u+v) = au + av for all a, u, and v
8) 1v = v for the number 1 in K

He says in a footnote that another mathematician informed him that (1) can be derived from (2) thru (8).

Here is an outline of what I did:

First, the zero vector is unique. Let O and O' be two zero vectors. Then O+O = O. But by (4), there is a vector "a" such that O + a = O'. Then

O + O + a = O + a
O + O' = O'
O = O' since O' is assumed to be a zero vector

The next useful fact is that 0v = O for all v. From (8) we have 1v = v, so

(0+1)v = v
0v + 1v = v
0v + v = v
0v + v + v' = v + v'
0v + O = O
0v = O

This also implies that for all v, O + v = v.

And then we need that for any v, its opposite v' is unique and is equal to -1v. From the preceding,

0v = O
(-1 + 1)v = O
-1v + 1v = O
-1v + v = O
-1v + v + v' = O + v' = v'
-1v + O = v'
-1v = v'

Now for any u,v, consider the vector w = u + v. Its opposite is w' = -1(u+v). But the vector w" = -1(v + u) is also an opposite for w since

w + w" = (u + v) + -1(v + u)
= (u+v) + -1v + -1u
= (u+v) + v' + u'
= u + (v + v') + u'
= u + O + u'
= u + u' = O

Since opposites are unique, -1(u + v) = -1(v + u), and multiplying by -1 gives
u+v = v+u.

 The only complaint I have of Sharipov is that there are no exercises included.
 Recognitions: Homework Help Science Advisor well when you learn to elarn you realize that every statem,ent in a good book is a siource of exercises. you want to start practicing making them up. nice proof hackab, much nicer than mine!

 Quote by mathwonk well when you learn to elarn you realize that every statem,ent in a good book is a siource of exercises. you want to start practicing making them up.
i just realised that myself, at last! & because of that i'm finally starting to understand things now. all i do is look at the definitions & the statements of the theorems. as i go along i can 'predict' with some accuracy what theorems will come up but i still don't totally make up theorems myself & try to prove them yet. i guess that's the next step.

 which site for vector space
 Recognitions: Homework Help Science Advisor congratulations fourier jr! notice how when you do that, even if you don't figure out the prroof yourself, when you read it you see that you did get maybe half of it, and it makes the other half look easier. so it focuses your attention on only that part of the exposition that you did not quite figure out yourself.
 My linear algebra teacher was awesome, he swore often and kept saying how much he hates arithmetic&numbers. As well as doing proofs for everything.
 Recognitions: Gold Member Homework Help Science Advisor Swearing often sure is a big plus.

 Quote by quasar987 Swearing often sure is a big plus.
Yes it is. It sort of relieves the tension in the room. Although it is possible to overdo it, too.

One of my linear algebra teachers (we had different teachers for different semesters) was not so awesome. He was determined to fill up the lecture period with worked numerical examples, but he always screwed up the numbers. That made taking notes useless. To this day, I am weak at "basic" stuff like gaussian elimination and computing determinants, cramers rule, etc. I should probably do something about that, since I think it will help me learn abstract linear algebra better. Sometimes proofs rely on these "elementary results" that the reader should "recall from an introductory course."

 Recognitions: Homework Help Science Advisor you only need to know that gaussian elimination is possible and how it is done. you do not at all need much skill atc arying it out, that is what calculators do better than humans.

 notice (sighh...) that sheldon axler's "linear algebra done right" sells for about 1/3 the price of books on linear algebra done wrong.
I can't tell, is this sarcasm? I'm rather interested in your opinion on Axler's book since I'm currently reading it.

 Recognitions: Homework Help Science Advisor i have not read it. my point is that book prices are totally unrelated to qualityu of the book, but only to popularity of the book. hence the worst books are usually the most expensive. in a very few cases the best books are also popular such as courants calculus book, and apostols calculus book. in linear algebra the expensive books by strang, and shifrin - adams, and hoffman - kunze, are good, but there are excellent books that are much cheaper, even free. anyone arrogant enough to call his book "....done right" probably means the theory is there and in its proper place. at the very least it probably means he is trying to do it rioght, which most authors do not even pretend to do. A book like that is going to mercilessly criticized if it does not satisfy the rigorous expectattions its title raises. but you have to read it of course to know. in my case the title alone tells me i will not be able to use it at my school, in all likelihood.
 Recognitions: Homework Help Science Advisor TomMe, some reasons i do proofs myself and like them are: 1) i make a lot of mistakes and porving things saves me from this. 2) i get more mileage out of a statement if i know why it is true, since that tells me when to use it, and knowing when toi use a theorem is more important than knowing how to make a calculation. so for me proofs are a safety net and a users guide to the subject.
 Don't get me wrong, I have nothing against proofs. If anything, I like to see everything proven also. It's just that I don't see how going over theorem after theorem in class is giving the student real insight into the subject, especially during the first year. How often I just found myself copying what's on the black board without really paying attention. Maybe it's because I'm a physics student and not mathematics that I don't appreciate the approach. Or maybe I'm just a bit slow. That's part of the reason I'm studying on my own now. Anyway, if I were to become a teacher later on, I would leave most of the proofs to the textbook and would make sure to give the student the general idea of what he's learning and why.
 Recognitions: Homework Help Science Advisor well proofs are important but intuition is even moreso, and i have found that physics students have that intuition in abundance. that is why witten's ideasare such a huge boost to mathematics this past decade or so.