Linear and Abstract Algebra textbooks

[TomMe]

Hm, can't really imagine someone bringing algebra in an entertaining way. I thought Strang did a good job though.
I've had 2 different linear algebra teachers so far (long story), now THEY were boring. Proof after theorem after proof..you get the picture. Maybe it's because the educational system is a bit different here, because most teacher's I've had were boring..

 

[mathwonk]

maybe its because you do not appreciate mathematics taught with proofs.

 

[HackaB]

Okay, I'll repeat my question from post #18 since I'm still curious. Does anyone else find it surprising that you can prove that vector addition is commutative using the other 7 axioms that Sharipov gives (they seem to be the standard ones)? Or is this old hat? It was news to me.

edit: FWIW, I thought this question was relevant to this thread because mathwonk posted the link to Sharipov's book here.

 

[mathwonk]

how does the proof go?

lets see:

2u + 2v = 2(u+v) = (u+v)+(u+v) = u + (v+u) + v

so subtracting v from the right gives, 2u + v = u + (v+u).

i.e. u + u+v = u + v+u.

now subtract u from the left.

?

 

[HackaB]

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.

 

[Icebreaker]

The only complaint I have of Sharipov is that there are no exercises included.

 

[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.

nice proof hackab, much nicer than mine!

 

[fourier jr]

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 realized 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.

 

[heman]

which site for vector space

 

[mathwonk]

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.

 

[gazzo]

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.

 

[quasar987]

Swearing often sure is a big plus.

 

[HackaB]

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."

 

[mathwonk]

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.

 

[rachmaninoff]

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.

 

[mathwonk]

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.

 

[mathwonk]

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.

 

[TomMe]

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.

 

[mathwonk]

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.