Help with Quotient Spaces: A Guide

[matheinste]

Help needed to eventually understand Quotient spaces.

 

[StatusX]

The notation v+W refers to what is called the coset of W containing v, which is just the set:

$$v + W = \{ v + w | w \in W \}$$

So, for example, 0+W is just W, as is w+W for any w in W, and v+W = v'+W (as sets) if and only if v-v' is in W (you should try proving these facts). So another way of thinking of the quotient space is as the set of cosets v+W. Scalar multiplication is defined by a(v+W)=(av)+W, and similarly for vector addition, and its not hard to check this makes V/W into a vector space with W as the zero vector.

 

[matheinste]

Thankyou StatusX.

I have looked at cosets and now understand ( or at least I think I do ) what is going on. My interest is in physics and so I will now go on to learn why we would construct such an entity as a quotient set in real situations. I will probably need more help in future.

Thanks again for your straightforward explanation. Matheinste.

 

[matt grime]

Quotient spaces occur when you want to set somethings equal to zero. That is where they occur through out maths in various context. In physics, you might want to quotient out some space of operators by those vanishing on certain initial conditions, for instance. Everytime you use vector spaces, you will use quotient spaces. There is nothing hard or mysterious about them. They are very good to think about because they force you to stop thinking about bases which are a hindrance to understanding anything.

 

[matheinste]

Hello Matt Grime.

I think I am heading in the right direction. I am getting the idea that by quotienting or factoring out the various ( infinite ) combinations of components, in various ( infinite ) bases that can be used to represent a geometrical vector you are somehow getting to the intrinsic nature of the vector itself. Is this somewhere close.

Matheinste.

 

[mathwonk]

cosets also arise from linear maps. a linear map from V to W defines cosets as follows: if K is the subspace of vectors mapped o zero, then the cosets iof zk are the sets of vectors maped to the same point. i.e. the inverse image of each point in the image is a coset of K.

 

[sp_math_stud]

Quocient spaces

Hi! I'm new in this, and I was looking for some people to help me in these things. I'm studying maths, and I had broken my head with this demonstration. In the quocient spaces, we demostrate that all vectors are independents and we study if they generate the quocient space. My problem with this is how I prove that one vector of this space (x) generate all the space.

Thanksfully,

Fatima

 

[HallsofIvy]

An example you may be familiar with is "modulo arithmetic". The set of all integers is a group and the set of all multiples of 3 is a subgroup. It's cosets are {0, 3, -3, 6, -6, ...} (multiples of 3), {1, -2, 4, -5, ...} (numbers one larger than a multiple of 3) and {2, -1, 5, -4, ...} (numbers one less than a multiple of 3). Treating those cosets as objects them selves, we get a 3 member quotient group- the integers modulo 3.

 

[mrandersdk]

you can also use quotients in describing periodic boundary conditions in a very mathematical way (maybe not the way it is usually done). fx

$$\mathbb{R}/\mathbb{Z}$$

makes all numbers on the real axis that differ by a integer multple equal, so if you got a potential that has periodicity of 1 then it is actully a function

$$f: \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}$$

like when using te first Brillouin zone. guess that you can construct more difficult zones in 3D, if you make some clever quotient of R^3.