Help Needed: Abstract Algebra Textbooks & Dual/Quotient Vector Spaces

[SeReNiTy]

Lately I've been taking a unit that deals with abstract algebra and I'm finding myself not understanding the lectures at all. To make matters worse the unit doesn't have a reccomended textbook so I don't even have any infomation to self learn from.

I guess what I'm asking is for some good reccomendations to textbooks for someone who is new to this field or could someone at least explain the concept of dual and quotient vector spaces?

Cheers, Serenity

 

[JasonRox]

Believe it or not, you should try just searching for it online. Just search for the definitions and examples.

That's what I do lots of times. It's easier than going through a textbook sometimes.

Try searching on Google for your questions. Universities post lecture notes everywhere, and other website simply have the information for you along with examples.

I could recommend a textbook, but I don't know much about your background because I don't want to recommend a textbook that's too difficult or too easy.

 

[Euclid]

My personal favorite source for quick info:
http://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)

 

[matt grime]

Let V be a vector space over F, V* is the linear maps from V to F. You know that the set of linear maps from an n-d space (written as columns vectors) to a 1-d space is just row vectors of length n (acting on the left). So that's all a dual space is. Given any element in V there is an obvious element in V* by just taking the transpose of the column to get a row.

Quotient spaces are much nicer because you can't just pick a basis, so you actually need to think about vector spaces properly.

Let W<V define an equivalence relation on V by x~y iff x-y is in W. Then the set of equivalence classes of V/~ carries a vector space structure.

Why do it this way? Well, the point is that there is no canonical way to pick complementary subspaces to W<V, but obviously any two complements you pick are isomorphic, and we're just writing down that information.

If we were in the plane with basis i,j, say, and we took the subspace to be that generated by i (the x axis), then there is an obvious choice of complementary subspace, that spanned by j (ie the y axis), but there are infinitely many other choices I can make: any line through the origin not equal to the x-axis for instance, and no one of them is preferred.

But, if I pick any two different complementary subspaces A and B, I can write any point as a combination

p=a+ti=b+si for some choice of vectors a in A, b in B and scalars t and s. Right? Notice that a-b differs by some element (t-s)i, so in the quotient they are the same.

So, where is it you start to get lost?

 

[SeReNiTy]

I'm understanding dual spaces now, but quotient spaces I am still completely lost. I don't know what the ~ symbol means and what it means to be a quotient space, it doesn't seem to be a easy concept.

A little history about the unit I am studying, its geometric aspects of general relativity, i think the aim is to build the mathematical structure of GR. Unfortunately i haven't taken many of the pre-reqs such as linear algebra or tensor study so I am lost a lot of the time...

 

[SeReNiTy]

I'm also confused about proofs, can somebody provide me with a proof for why general functions form a vector space. I know you have to prove the axioms but i have no idea on how to write this out formally.

 

[HallsofIvy]

I'm also confused about proofs, can somebody provide me with a proof for why general functions form a vector space. I know you have to prove the axioms but i have no idea on how to write this out formally.

"General Functions"?? Do you mean just any function from a field F to itself?
Define f+ g by (f+g)(x)= f(x)+ g(x) and define af (for a a member of field F) by (af)(x)= a(f(x)). Those are both defined since addition and multiplication are defined in a field. In fact, they are clearly associative, commutative, etc. because addition and multiplication in the field are.
You need, then, to show that f(x)= 0 for all x is the additive identity (0) for the vector space and, for each f, g(x)= -f(x) is the additive inverse. That's about all there is to it. The set of all functions from a field F to itself is a vector space over F.

 

[SeReNiTy]

"General Functions"?? Do you mean just any function from a field F to itself?
Define f+ g by (f+g)(x)= f(x)+ g(x) and define af (for a a member of field F) by (af)(x)= a(f(x)). Those are both defined since addition and multiplication are defined in a field. In fact, they are clearly associative, commutative, etc. because addition and multiplication in the field are.
You need, then, to show that f(x)= 0 for all x is the additive identity (0) for the vector space and, for each f, g(x)= -f(x) is the additive inverse. That's about all there is to it. The set of all functions from a field F to itself is a vector space over F.

Yes I've done that and reconized the properties the vector space will need to satisfy. However, i don't know how to present this with the formal notation that my professor expects, could someone provide a full proof so i can learn how to do all vector space proofs?

I've also heard of a method of subspace, so find a vector space that is a subspace of a known vector space. That way we don't need to prove all axioms, just closure conditions. This seems confusing since how do you formally prove closure?

 

[matt grime]

There is no method that proves everything for any case of any thing. You just have to do it. How would you prove something somewhere other than maths? To show closure you need to show that if you add up two elements satisfying some condition you get another satisfying the same condition. How you do that depends on the space and the condition.

I'm also confused about proofs, can somebody provide me with a proof for why general functions form a vector space.

they don't so you cannot, but as was pointed out you probably haven't stated the question fully.

I don't know what the ~ symbol means and what it means to be a quotient space, it doesn't seem to be a easy concept.

it is easy if you know what an equivalence relation is. Try finding out about these before looking at quotient spaces. If you understand modulo arithmetic you can understand this, and you use modulo arithmetic all the time: every time you work out what day it is in 12 days time you're doing modulo arithmetic. The day in 12 days time is the same day as it is in 5 days time, because we're declaring dates that differ by 7 to be the same day. And the day 5 days hence is the same day as two days ago, thus it will be a Monday (for me at the time of writing this). You're doing the same thing here but you just don't have nice labels for things like monday, wednesday etc. Essentially all you need are examples.

Consider the real numbers R, and consider V to be the set of functions from R to R. This is a vector space over R: the sum of two functions from R to R is a function from R to R, as a scalar multiples of of functions from R to R. the function f(x)=0 is a zero vector.

The set of functions W satisfying g(0)=0 is a sub vector space: if g and h vanish at 0 so does rg+th for any r,t n R.

It has a quotient space V/W. It is the set of all functions from R to R except that whenever we see two functions p(x) and q(x) such that p(0)-q(0)=0 (or p(0)=q(0)) then we'll say they represent the same vector, just like dates separated by 7 days have the same name.

Let's make it even easier to visualize. Suppose V is actually the set of all polynomial functions from R to R, and W is defined accordingly, then V is a vector space, W a subspace and V/W can be explicitly described. p(0) is just the constant term! So we're saying that the quotient is where we identify polynomials with the same constant term. We can get an explicit descrption of this since we can pick a distinguished name for all polynomials with the same constant term, if the constant term is k in R, let the name also be k (this is just like naming things monday or tuesday), so the quotient space is exactly R as a vector space.

 

[Nolen Ryba]

This isn't exactly rigorous but it may help you understand the idea of a quotient space in terms of stuff you already know. Suppose you have an inhomogenious differential equation that can be written as Lu=f where L is some operator, and u and f funtions. To solve this you are taught in a first course to first solve Lu=0, which gives you the complimentary soluton, a few functions multiplied by arbitrary constants, and then find a particular solution. The total solution is then the sum of the complimentary solutions and the particular solution. The key here is that you shove the particular solution into the equation and it maps to f, and you can add any combination of the functions of the complimentary solution to it and it still maps to f because L is linear and the complimentary solution always maps to 0. What you are essentially doing here is finding the element of the quotient space H/(complimentary function space) that maps to f. The complimentary solution is typically expressed as a linear combination of elements a basis of the nullspace of the operator L. The quotient space is used because it's nullspace contains only the zero element (by construction) and so the map is invertible. If you were to use the original space of functions, multiple functions map to the same f, namely any function that maps to f + any element of the nullspace. I wish I could find a good picture on the web of this (set diagram type of idea)...it always seems to make things more intuitive. You might want to look at group homomorphisms and quotient groups to make more sense of the formalities, but I personally find the linear algebra ideas more intuitive.

 

[Nolen Ryba]

woops...forgot to define H- whatever function space you are working in