# If the Quotient Space contains $0+U$ as it's zero element, must we not then include it in the visualization?

In Axler's book Linear Algebra Done Right 3.ed he defines the quotient space to be $$V/U=\{v+U:v\in V\}$$. As an example it is stated, that if $$U=\{(x,2x)\in\mathbb{R}^2:x\in\mathbb{R}\}$$ then $$\mathbb{R}^2/U$$ will consist of all lines with slope 2.

This makes sense to me. As we have $$x_0=(0,0)\in\mathbb{R}^2$$, then according to the definition we should have $$x_0+U\in\mathbb{R}^2/U$$, which means that $$U$$ itself is included in the quotient space $$\mathbb{R}^2/U$$ and also functions as the additive neutral element in the vector space $$\mathbb{R}^2/U$$.

In the next example I get confused because it is stated that:

If $$U$$ is a plane in $$\mathbb{R}^3$$ containing the origin, then $$\mathbb{R}^3/U$$ is the set of all planes in $$\mathbb{R}^3$$ parallel to $$U$$.

Does this mean that $$U$$ is in $$\mathbb{R}^3$$ or not? It seems to me that the definition of affine subsets and parallel allows for $$U$$ to be parallel to itself and thus belonging to $$\mathbb{R}^3$$/U.

And if $$U$$ belongs to any quotient space $$V/U$$, then I think I have to adjust my visualization of quotient spaces. Up until now, I have visualized a quotient space as $$V$$ with $$U$$ filtered out, so that $$V/U$$ can be seen as the complement of $$U$$. However, if $$U$$ is included in the quotient space then I believe it would be more appropriate to see $$V/U$$ as $$V$$ being partitioned into chunks of $$U$$.

• If you want to visualize $V/U$, it is just the orthogonal complement of $U$ whenever there is a notion of inner product. Whereas, in abstract sense, $V/U$ is disjoint union of affine subspaces of $V$. Since $U$ is a subspace of $V$, it is affine as well, and hence $U \subset V/U$.
– MAS
Apr 26, 2022 at 14:02