What the frick are quotient (vector) spaces in linear algebra? How can they be pictured and what is their significance?
What the fuck are quotient (vector) spaces in linear algebra? How can they be pictured and what is their significance?
What the frick are quotient (vector) spaces in linear algebra? How can they be pictured and what is their significance?
it rquals 12000
you will never have enough angular momentum
At a certain point what's the point of visualization? You'll never see it in experimental physics or anything so just follow the logic and move on imo, it's all logical circle jerking eventually
V/W => subspace of V normal to W
How do you arrive at this conclusion? The definition uses equivalence classes so you don't deal with the vectors themselves right?
Yes but you are dealing with colections of vector spaces
According to wikipedia
> Alternatively phrased, the quotient spaceV/Nis the set of allaffine subsetsofV which areparallel toN
What if the space doesn't have an inner product?
All spaces have an inner product.
Based highschool chad dabbing on topological vector spaces
If the space is a direct sum, [math]V oplus W[/math], quotienting out by V gives W.
This generalizes the case of an orthogonal complement, [math]mathbb{R}^n = V oplus V^{perp}[/math].
If the space is not a direct sum?
Then the quotient=orthogonal complement interpretation doesn't work.
The general formula is
[eqn] (V + W)/V simeq W/(V cap W)[/eqn]
If [math] V + W [/math] is a direct sum then [math]V cap W = {0}[/math] and
[eqn]W/(V cap W) simeq W [/eqn]
so
[eqn] (V oplus W)/V simeq W[/eqn]
You really don't need to bring an inner product in here.
That symbol indicates the difference. In latex, it's called "setminus." One way to visualize this "quotient" is
[math]mathbb{C}setminusmathbb{R}=(-iinfty,iinfty)[/math].
Even though people call it the quotient (sometimes) it's a difference.
>V/W => subspace of V normal to W
This can also be stated as
V/W => V minus W
You can see that (-i*INF, i*INF) is the "subspace of C normal to R. If one can't see that, the setminus symbol is probably not a learning that they should prioritize.
sorry if I am misunderstanding something, but the symbol / does not indicate difference. the quotient of two algebraic objects is not in any way equal to their set difference. [math] mathbb C setminus mathbb R[/math] is not equal to [math] (-iinfty, iinfty) [/math] either.
The namegay you're responded to is a total crank who doesn't know basic math. That's the misunderstanding.
Geometrically you're just collapsing the subspace to a single point.
Wrong, idiot.
I always visualize them as a big land field divided into small family plots. Many such divisions are possible. Of course im talking about lie group quotients. Vector spaces? Probably just a type of subspace
My confusion is that a subspace is a subset of a vector space (so just a set of vectors) while the quotient space is technically a set of subspaces (by construction with equivalence classes). However the quotient space is treated like a subspace, since it is itself a linear vector space. I guess the elements of the quotient space can be identified somehow with a certain subspace of the vector space. It seems weird to me why someone would define them that way, through equivalence classes that is.
Equivalence classes are usually the wrong way to think about quotients. They're basically just a detail of how they're encoded into set theory.
You should think of V/W as the same thing as V, but where W has been collapsed to 0. Specifically, it's V, but with "more equalities": w=0 now holds for any w in W, plus any consequences of that. You could also say it's V, but ignoring difference by W.
For vector spaces, every vector can be written as a sum w + u where w is in W and u is normal to W. When you quotient out by W, you set w=0 so only the part normal to W is left.
A quotient is the image of the canonical morphism induced by a congruence (an equivalence relation which is also a subalgebra of the direct square). Hope this helps!
Basically partitions the space into equivalence classes where the equivalence relation is membership relation of subspaces and two elements are equivalent if their difference is a member of the subspace that is being used to construct the quotient. Also the elements in the subspace W(e.q. V/W) are collapsed to being trivial in the quotient so they act kind of as a zero vector in the original space. Also the canonical map sending each vector to its equivalence class is surjective. This is a purely algebraic definition which is quite suitable for wide use. The example you give in the pic. rel. collapses the boundary of the n-dimensional disk to the trivial class so the quotient space is isomorphic to the n-dimensional sphere.
correction: homeomorphic not isomorphic
a homeomorphism is an isomorphism of topological spaces
I know but the definition of a homeomorphism includes a continuity. condition while the pure isomorphism semantic in the category Set doesn't so I wanted to be more precise. But yeah a homemorphism is an isomorphism in the Top category.
correct but not pedagogical so useless when the point is helping a confused person
>freshmen can't tell the difference between set difference and quotient space
ngmi
That guy isn't a freshman he's just a schizo who thinks he solved the Riemann hypothesis.
I wasn't paying attention to the name
Algebraic topology (which is where I work with quotient spaces nonstop) is some of the most beautiful shit I've ever seen, so I'll try to help out. I don’t know your level so I may be too pedantic or too advanced at times.
In finite dimensional linear algebra, I never really needed to worry about quotient vector spaces. The idea of a quotient space is that you want to study an algebraic structure by studying what happens when you collapse a part of it.
For example, in groups (if you don’t know what it is, look it up on wikipedia. the definitions should be straight-forward at least), the idea of a quotient group arises by taking a “normal” subgroup and looking at all the “cosets”. This imo seemed very contrived and hard to work with at first, but it is actually a very nice idea.
Consider integers [math] mathbb Z [/math]. The set of even integers is a subgroup of integers because whenever you add two evens you get an even. The quotient subgroup [math] mathbb Z/2mathbb Z[/math] (read “Z mod 2Z”) is the set of all “cosets” [math] n + 2mathbb Z = {n + 2mmid min mathbb Z}[/math] under the operation [math] (n + 2mathbb Z) + (n’ + 2mathbb Z) = (n + n’) + 2mathbb Z [/math]. We also impose the restriction that [math] (n + 2mathbb Z) = (n’ + 2mathbb Z) [/math] if, and only if, [math] n - n’ in 2mathbb Z[/math]. This ends up being isomorphic as groups (a bijective group homomorphism, basically they have the same group structure) as addition on the integers modulo 2.
In the case of finite dimensional vector spaces, you are doing a very similar thing. We have something like [math] mathbb R^3 [/math] and we know that [math] mathbb R^2 [/math] exists inside of it. We can look at all “cosets” of the form [math] x + mathbb R^2 [/math]. These cosets are planar subsets of [math] mathbb R^3 [/math]. If you look at every plane perpendicular to the z-axis, with the operation that adding two planes adds their height to get a new plane, and scaling a plane by lambda scales it to a new plane with its old height times lambda, what we have a is a one-dimensional vector space. This seems almost useless, but the idea is very important in general, and it is at the very least a good kind of intuition to have.
In the case of topological spaces, your best bet is to just think about gluing points together. There is a really nasty definition of quotient space that is usually not worth worrying about. The idea is you take a topological space, like [math] D^2 [/math], the closed 2-dimensional disk, and then you “glue” every single point on the boundary together. It’s kind of like if you drew a path from one point to a boundary point, you could keep drawing from any boundary point you like. The topology on this space is basically the old topology except now every open set of a point on the boundary must contain every single boundary point. It is nice to visualize as a circular sheet of (stretchy) paper where you pull all of the edges up to turn it into a balloon. A balloon is basically a sphere by topological standards, so the quotient space is homeomorphic to [math] S^2 [/math].
This is an actual good thread. It represents the kind of discussion I come to IQfy for
I think you ought to understand more general quotient spaces first. Take [math] mathbb{Z} / 5mathbb{Z} [/math] which itself is a quotient space. Here we just see the modified equivalence to the new congruence that the two numbers have a difference divisible by 5.
With Linear algebra, it is the same idea. We collapse the entire space into a new space where any differences which lie in the "divisor" space are taken to be zero.
Just as the case with modulo arithmetic being useful in throwing irrelevant information away, the same holds for quotient spaces. You can effectively ignore how vectors from that space contribute to the final result. A lot of times, you can use them for symmetry to allow you to treat translation by vectors of a certain type as if they don't have an impact.