Understanding Cosets and Subspaces in Linear Algebra

In summary: 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".
  • #1
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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.
 
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  • #2
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.
 
  • #3
Check out http://en.wikipedia.org/wiki/Quotient_vector_space" [Broken].

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 [itex] v \in V[/itex] and [itex]v+W = \{v + w : w \in W \}[/itex]. A coset v+W is not a subspace itself unless [itex] v \in W [/itex]. Try showing [itex] v + W [/itex] is a subspace if and only if [itex] v \in W [/itex].
HallsofIvy said:
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?
 
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  • #4
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,...,wk,v1,...,vn-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.
 
  • #5


Hello,

Cosets and subspaces are both important concepts in linear algebra, but they have different definitions and properties.

A coset is a set of elements that are obtained by adding a fixed element to all elements in a specific subset of a vector space. In other words, a coset is a translated version of a subset of a vector space. It is denoted by a + W, where a is a fixed element and W is a subset of the vector space.

On the other hand, a subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. This means that if you add or multiply any two vectors in the subspace, the result will also be in the subspace. Subspaces are denoted by W.

To summarize, cosets are translated versions of subsets, while subspaces are subsets that have specific properties. Cosets and subspaces can have different dimensions and may not always be equal. It is important to understand the differences between these concepts in order to solve problems in linear algebra effectively.

I hope this helps clarify the difference between cosets and subspaces. Keep practicing and exploring, and you will continue to build your understanding of linear algebra. Best of luck on your journey!
 

1. What is a coset in linear algebra?

A coset is a subset of a vector space that is formed by adding a fixed vector to every vector in a subspace. It is essentially a translated version of the subspace and shares many of the same properties and operations as the original subspace.

2. How are cosets related to subspaces?

Subspaces and cosets are closely related because cosets are formed by adding a fixed vector to every vector in a subspace. In fact, every coset is a subset of the vector space that contains the subspace, and the subspace itself is a coset of itself.

3. What are the key properties of cosets?

There are several key properties of cosets, including closure (the sum of two cosets is also a coset), associativity (the order of adding vectors to a coset does not matter), and the fact that cosets partition the vector space (every vector in the vector space belongs to exactly one coset).

4. How are cosets used in linear algebra?

Cosets are used in linear algebra to help understand the structure of a vector space and its subspaces. They can also be used to solve systems of linear equations and to perform operations such as finding the inverse of a matrix.

5. Can cosets be used to prove theorems in linear algebra?

Yes, cosets can be used to prove theorems in linear algebra. For example, the Fundamental Theorem of Linear Algebra can be proven using cosets. Cosets can also be used to prove theorems related to vector spaces, subspaces, and linear transformations.

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