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Consider a finite-dimensional vector space, $V$ over a generic field $F$.
Suppose $v\in V$ and $U$ is a subspace of $V$. Define an affine subset of $V$ as:

$v+U=\{v+u:u\in U\}$

$\textbf{My Question:}$ I am little confused about the affine subset of a vector space. So, is an affine subset $always$ parallel to $U$? The way the below reference defines the quotient space is that it is the set of all affine subsets of $V$ parallel to $U$. Isn't this tautology? In other words, is there a set of affine subsets of $V$ $not$ parallel to U?

Reference: Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.

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3 Answers 3

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Well, if you define parallel as being disjoint, then you're right. If you consider a subspace $U$ and an affine subspace $V=v+U$, $v\in W$, of a vector space $W$, then $U\cap V=\emptyset$ iff $v\in W\setminus U$. Indeed, if $u\in U$ lies also in $V$, then $u = v + u'$ for some $u'\in U$ and so $v=u-u'\in U$.

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  • $\begingroup$ Thank you for your response! :) $\endgroup$ Jun 27, 2019 at 7:42
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Let $u, v\in V$ be colinear but not of the same length, that is $u=av$ for some $a\in F$. Then $v+U$ and $u+U$ are parallel (and both are parallel to $U$) but they're not the same affine subspace.

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  • $\begingroup$ Thank you for your response! :) $\endgroup$ Jun 27, 2019 at 7:42
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We call $v+U$ an affine subset for arbitrary subspace $U$ and vector $v$.
So, yes, every affine set is parallel to some subspace.

An alternative description is that a subset $W$ is affine iff it is closed under affine combinations, i.e. for all scalars $\alpha_1,\dots,\alpha_n$ with $\sum_i\alpha_i=1$ and vectors $w_1,\dots, w_n\in W$, we have $$\sum_i\alpha_iw_i \in W$$

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  • $\begingroup$ Thank you for your response! :) $\endgroup$ Jun 27, 2019 at 7:42

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