Understanding Vector Quotient Spaces in Linear Algebra

In summary, We discussed the concept of Vector Quotient Spaces, which involves taking a vector space $V$ and quotienting it by a linear subspace $N$. This results in the set $V/N$ which consists of equivalence classes $[N + v]$ where $v \in V$. An example was given of taking $\mathbb{R}^{2}$ and quotienting it by the x-axis, resulting in a set of points that are a distance r away from the x-axis. This can be rewritten as y=r. There was some difficulty in understanding how cosets $x-axis + r$ could yield lines parallel to the x-axis, but it was clarified that by adding a vector to the x-axis,
  • #1
mruncleramos
49
0
I'm having a bit of trouble seeing Vector Quotient Spaces.
Lets say I have a vector space $V$ and I want to quotient out by a linear subspace $N$. Then $V/N$ is the set of all equivalence classes $[N + v]$ where $v \in V$.

For example, let me try to take $\mathbb{R}^{2} /$ x-axis. This should be the set of all equivalence classes $[x-axis + r]$ where $r \in \mathbb{R}^{2}$.

Here is where the difficulty arises I believe. I am told that this set is the class of lines parallel to the x-axis, but I can't see how any coset $x-axis + r$ could yield a line parallel to the x-axis - or maybe my conception of vector space cosets are wrong.
 
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  • #2
If you have a class that is the set of points a distance r away from the x-axis, you can rewrite it as y=r.

Try graphing it and see for yourself (pick an r, say, r=1)
 
  • #3
I can't see how any coset $x-axis + r$ could yield a line parallel to the x-axis
What points lie in $x-axis + r$?
 
  • #4
But if you add a vector to the x-axis, how can that be a line parallel to the x-axis. More importantly, what is an element of the x-axis? Is it just a point?
 
  • #5
More importantly, what is an element of the x-axis?
Since the x-axis is a vector space... a vector!

(Of course, sometimes we identify the notions of "point" and "vector" with each other)
 
  • #6
Then i think my conception of adding vectors is incorrect. If i think of vectors as directed line segments, i can't think of a way to add vectors on the x-axis to other vectors to get something parallel.
 
  • #7
You're not. What you're doing, is adding a vector to the x-axis, and the set of ends of those directed line segments forms a line.
 
  • #8
oh oh oh i see now. thanks
 
  • #9
and the line comes from the vector definition of line.
 

1. What is a vector quotient space?

A vector quotient space is a mathematical concept that combines elements of vector spaces and quotient spaces. It is a vector space created by dividing a vector space by a subspace, resulting in a new space with modified operations and elements.

2. How is a vector quotient space different from a regular vector space?

In a regular vector space, operations like addition and scalar multiplication are performed on elements within the same space. In a vector quotient space, these operations are performed on elements that have been "quotiented" or divided by a subspace, resulting in a new set of operations and elements.

3. What are some real-world applications of vector quotient spaces?

Vector quotient spaces are often used in physics and engineering, particularly in fields such as mechanics and electromagnetism. They are also used in computer graphics and computer vision to represent and manipulate geometric objects.

4. How are vector quotient spaces related to linear algebra?

Vector quotient spaces are a branch of linear algebra that deals with vector spaces and their subspaces. They offer a way to study and manipulate vector spaces with additional structures and properties, making them useful in various mathematical and scientific applications.

5. Are there any limitations or drawbacks to using vector quotient spaces?

One limitation of vector quotient spaces is that they can be quite abstract and difficult to visualize. They also require a solid understanding of linear algebra and quotient spaces, which can be challenging for some individuals. Additionally, the operations and properties of vector quotient spaces may not always translate well to real-world scenarios.

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