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Definition quotient space, $V/U$

Suppose $U$ is a subspace of $V$. Then the quotient space $V/U$ is the set of all affine subsets of $V$ parallel to $U$. In other words, $V/U = \{v+U: v\in V\}$.

Then 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 $U$. The dimension of this quotient space should be $2$.

But let $\pi$ be the quotient map from $V$ to $V/U$. $\text{null } \pi = U$, $\text{range }\pi = V/U$ and $\text{dim } V = \text{dim } U + \text{dim } V/U$.

Then if $U$ is a plane in $\mathbb R^3$ containing the origin, the dimension of $\mathbb R^3/U$ should be $3-2=1$.

What is wrong with my thoughts about the dimension of quotient space?

Thanks

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    $\begingroup$ What is the justification for your statement "The dimension of this quotient space should be 2", after giving the definition? Everything else you wrote seems correct. $\endgroup$ Jan 9, 2023 at 14:36
  • $\begingroup$ Because they are planes in $\mathbb R^3$ $\endgroup$
    – Dan Li
    Jan 9, 2023 at 14:40
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    $\begingroup$ They are indeed planes, but the space of all these planes can be parametrized by a single number, since you already know that they are all parallel. Does this help? $\endgroup$ Jan 9, 2023 at 14:41
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    $\begingroup$ Yes, I understand, all these planes can be parametrized by a line perpendicular to them. $\endgroup$
    – Dan Li
    Jan 9, 2023 at 14:44
  • $\begingroup$ Exactly! That should solve your problem. All the best $\endgroup$ Jan 9, 2023 at 14:51

1 Answer 1

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Your understanding of the definition of the quotient space is correct, but your application of the formula for the dimension of the quotient space is incorrect.

The formula for the dimension of the quotient space is indeed: dim V=dim U+dim V/U

However, this formula only holds when the quotient map is a linear map. In your example, the quotient map is not a linear map, since it takes an affine subset (a plane) to a point in the quotient space (the set of all planes parallel to U).

Therefore, the formula for the dimension of the quotient space does not apply in this case, and the dimension of the quotient space R3/U cannot be computed using this formula.

Instead, you can think of the quotient space R^3/U as a set of points, where each point represents a plane in R3 parallel to U. In this sense, the dimension of R^3/U is indeed 0 since each point in the quotient space represents a single plane, rather than a subspace of dimension greater than 0.

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