# Is Affine Subset always Parallel?

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.

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 $$v=u-u'\in U$$.

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

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.

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

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 $$\sum_i\alpha_iw_i \in W$$

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