Understanding Vector Quotient Spaces in Linear Algebra

[mruncleramos]

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.

 

[Office_Shredder]

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)

 

[Hurkyl]

I can't see how any coset $x-axis + r$ could yield a line parallel to the x-axis

What points lie in $x-axis + r$?

 

[mruncleramos]

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?

 

[Hurkyl]

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)

 

[mruncleramos]

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.

 

[Office_Shredder]

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.

 

[mruncleramos]

oh oh oh i see now. thanks

 

[mruncleramos]

and the line comes from the vector definition of line.