Understanding Cosets and Subspaces in Linear Algebra

[Anshuman_]

Hi,
I have just begin with Linear Algebra.

I came across cosets and I don't understand what is the difference between cosets and subspaces?

thanx in advance.

 

[HallsofIvy]

Where in the world did you run across "cosets" in Linear Algebra? Subspaces are a common topic in Linear Algebra but "cosets" are from group theory.

 

[spamiam]

Check out http://en.wikipedia.org/wiki/Quotient_vector_space" .

For a vector space V with a subspace W, a coset is a "translate" of W, i.e. something of the form v+W, where $v \in V$ and $v+W = \{v + w : w \in W \}$. A coset v+W is not a subspace itself unless $v \in W$. Try showing $v + W$ is a subspace if and only if $v \in W$.

Where in the world did you run across "cosets" in Linear Algebra? Subspaces are a common topic in Linear Algebra but "cosets" are from group theory.

What would you call the elements of a quotient vector space if not cosets? A vector space is an abelian group, so doesn't it make sense to call them cosets?

 

[Deveno]

although one can define subspaces and cosets abstractly, it is sometimes helpful to "see" what these things "are" in terms we can visualize.

so, for the purposes of this post, we will assume our "master vector space" is R³, which we can imagine as being very much like the space we live in.

spatially, all the information for a vector space, is encoded in a basis, which you can also think of as "a coordinate system". so dimension (the size of the basis) corresponds to "degrees of freedom of movement" how many coordinates you need to specify to say where something is. R³ has dimension 3, and the standard basis {(1,0,0),(0,1,0),(0,0,1)} is the usual "xyz"-coordinate system.

what is a subspace in R³? it is a vector space in R³ of equal or lesser dimension.

the 3-dimensional subspace of R³ is not that interesting, it's just R³.

2-dimensional subspaces ARE interesting. we call these "planes" (to be subspaces, they need to pass through the origin). if we have two linearly independent vectors (which in R³, means they aren't "on the same line") they determine a plane (every point in that plane is a linear combination of those two vectors).

1-dimensional subspaces are also interesting. these are all just scalar multiples of a single non-zero vector, and so are lines passing through the origin.

finally, the origin itself forms a rather dull vector space: {(0,0,0)}.

now...let's look at the cosets!

a coset of 0 = (0,0,0) is isn't very interesting at all, it's just an ordinary vector (x,y,z). cosets of R³ aren't very interesting, either, there's only one, R³ = (0,0,0) + R³
.
a coset of a line L = t(a,b,c) is interesting, this is just a line of the form (x,y,z) + t(a,b,c), parallel to (a,b,c) and going through (x,y,z). you can think of this as the line L moved from (0,0,0) to (x,y,z), so it makes sense to call it: (x,y,z) + L. what does the "coset space" look like?

the way i like to imagine it, is as a bundle of infinite straws (really, really thin ones), all in the same direction. to say "which" straw we're on, we imagine a plane cutting through all the straws, so it just takes 2 coordinates. for example, if our line is the z-axis:

L = (0,0,t) = t(0,0,1) (where t can be any real number), then all the straws point "straight up" and we have a corresponence:

(x,y) <---> (x,y,z) + L (we don't care about the z-coordinate, because changing it doesn't change "which straw we're talking about").

ok, how about a coset of a plane P = s(a,b,c) + t(u,v,w)?

i think of this like a deck of "infinitely big cards", or like sheets of plywood, all stacked together. to specify "which sheet" we're on, we pick a line going through all the sheets, and by specifying a point on that line, we know which sheet we live on.

again, suppose our plane was the xy-plane, P = (s,t,0) = s(1,0,0) + t(0,1,0). then we have a correspondence between:

z <---> (x,y,z) + P (we don't care where in the xy-plane we are, just how far "up or down").

so a coset of a line (1-dimensional subspace) is a 2-dimensional "bundle of lines", and a coset of a plane is a 1-dimensional "stack of planes". note the relationship:

space subspace coset space
dim n...dim k...dim(n-k)

so another way to think of a coset space (or quotient space) is:

"shrink k dimensions down to 0" or "those k degrees of freedom aren't relevant", we are essentially "identifying all the elements of W" are being "equivalent", so if:

{w₁,w₂,...,w_k,v₁,...,v_n-k} is our basis for V, we just "ignore" the "w-component" and use the "v-coordinates" when we talk about the vector u+W in V/W.