Computing Quotient Space: Practical Example with Vector Spaces and Subspaces

In summary, a quotient space is a collection of equivalence classes defined by a subspace V, where two vectors are considered equivalent if their difference is in V. An example of a quotient space is when U is R2 and V is the linear span of (1,0). The quotient space U/V is then the set of all (a,b)+V, which can be simplified to (0,b)+V. In general, for a finite dimensional U with basis v1,...,v[sup]m[/sup] and extended basis u1,...,un, the quotient space U/V is isomorphic to the linear span of u1,...,un. The quotient space is always a vector space over the same field as U and
  • #1
jamjar
10
0
Can anyone post me a clear example of how to compute the quotient space U/V from a vector space U and subspace V?
I've seen many formal definitions but I'm a little stuck on practical use.
I'm particularly interested in an example that shows how U and V being over the reals (for example) can result in the quotient being over a finite field.
 
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  • #2
The quotient space should always be over the same field as your original vector space. To 'counterprove' your desired example, if U/V is over a finite field, the field has characteristic p, which means that for some u not in V, p*u is in V. But V is a vector space.

An example of a quotient space... I think in R2 is the best example. U is R2 V is the linear span of (1,0). Then U/V is going to be the set of all (a,b) + V. But this is ((a,0) + (0,b)) + V (a,0) is in V for all a, so (a,0)+V = 0+V which implies that (a,b) + V = (0,b) + V

In general for a finite dimensional U, if V has basis v1,...,vm[/sub] and this is extended to a basis of U by adding u1... un Then if u is in U, [tex]u = \sum_{1}^{m}a_iv_i + \sum_{1}^{n}b_iu_i[/tex] where the a's and b's are in the field. Then any vi+V = 0+V so [tex]u+V = \sum_{1}^{n}b_iu_i + V[/tex] and it's easy to see that U/V is isomorphic to the linear span of u1,...,un
 
  • #3
A "quotient space" is always a collection of equivalence classes. Given a subspace V, we can define "u~ v" if and only if u- v is in V.

For a simple example, let U be R2, the set of all pairs (x, y) with the usual addition and multiplication and V the subspace {(x, 0)}. Two vectors are (x1, y1) and (x2, y2) are equivalent if and only if (x1,y1)- (x2,y2) is in (x1-x2, y1-y2) is in V which means y1=y2. So an equivalence class consists of all {(x,y0} where x is a variable and y0 is a specific number. As points, those are all the points on the line parallel to the y axis, y= y0. Of course, in order to make that a vector space you have to define an addition and scalar multiplication. The standard way of "adding" equivalence classes is to choose one member from each equivalence class. For example, there is an equivalence class consisting of all pairs of the form (x, 1) and another equivalence class consisting of all pairs of the form (x, 5). One "representative" of the first class is (3, 1) and a "representative" of the second is (4, 5). Their sum is (3, 1)+ (4, 5)= (7, 6). That is in the equivalence class of all pairs (x, 6) so {(x, 1)}+ {(x, 5)}= {(x, 6)}. To multiply by a scalar, you can do the same thing: a*{(u,y0)}= {( u, ay0)}. Of course, the pair (0, y0) is always in the equivalence class {(x, y0} so you can always use that pair and you can think of this quotient space being the space of points (0, y) which is a subspace of R2[/sub].

If you like, you can think of that as "collapsing" the line y= y0 to the single point (0,y0) so it's easy to see that quotient space is really the same as the subspace {(0,y)}.

Having said all that, I come to your last question! The definition of "quotient space" I have given- which I think is standard- clearly gives the quotient space as a vector spaced over the samefield as U and V. I can't think of any way of starting with U a vector space over R, say, and U/V being a vector space over a finite field.
 
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1. What is a quotient space example?

A quotient space example is a mathematical concept that involves partitioning a given space into smaller, disjoint subsets and then identifying these subsets as equivalent points. This process creates a new space, known as the quotient space, which preserves the essential structure of the original space.

2. What is the purpose of using quotient spaces?

The purpose of using quotient spaces is to simplify complex mathematical structures and make them easier to analyze. By identifying equivalent points and creating a new space, we can reduce the number of elements and make the structure more manageable.

3. How are quotient spaces used in real-world applications?

Quotient spaces have various applications in fields such as physics, engineering, and computer science. They are used to model and analyze complex systems, such as fluid dynamics, electrical circuits, and data compression algorithms.

4. Can you give an example of a quotient space?

One example of a quotient space is the set of real numbers, which can be partitioned into equivalence classes based on their decimal representation. The quotient space, in this case, would be the set of rational numbers, where all numbers with the same decimal representation are considered equivalent.

5. What are some properties of quotient spaces?

Quotient spaces inherit certain properties from the original space, such as topological, algebraic, and geometric properties. They also have unique properties, such as the dimension of the quotient space being equal to the dimension of the original space minus the dimension of the partitioning subsets.

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