I'm having a bit of trouble seeing Vector Quotient Spaces.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Vector Quotient Spaces

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