I am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me.

I cite:

A subset $B$ of a $\mathbb{R}$-affine space $A$ modelled on $V$ is an affine subspace if there is a subspace $U$ of $V$ with the property that $y−x \in U$ for every $x,y \in B$

It later says that this definition is equivalent to to the usual one, namely that of closeness under sum with elements of a $U$, but it seems to me that there is a problem with the first definition. Just imagine the usual $\mathbb{R}^2$ plane as an affine space modeled on $\mathbb{R}^2$. According to this definition the subset $\{(0,0);(0,1)\}$ is an affine subspace, while this is not so according to the usual definition of an affine subspace. Is there an error in the book?


The definition you cite is incorrect (so yes, there is an error). Indeed, letting $U = V$ every subset is an affine subspace according to this definition.

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  • $\begingroup$ By the way, for future reference this "modeled after" terminology is not standard in pure mathematics. One would say that an affine space is a torsor or principal homogeneous space for a vector space $V$. $\endgroup$ – Qiaochu Yuan Feb 24 '12 at 18:53

If you take a subspace and shift it away from the origin, you get an affine subspace.

In other words, an affine subspace is a set $a+U=\{a+u \;|\; u \in U \}$ for some subspace $U$.

Notice if you take two elements in $a+U$ say $a+u$ and $a+v$, then their difference lies in $U$: $(a+u)-(a+v)=u-v \in U$. [Your author's definition is almost equivalent to the one I've given above. The author mistakenly says "for all $x,y$ when it should be for any fixed $x$ all $y$ lie in there iff $x-y$ lie in the subspace.]

If you are familiar with a bit of modern algebra, affine subspaces are just elements of quotient vector spaces. So for example, given $U$ a subspace of $V$, the set $V/U = \{ a+U \;|\; a \in V\}$ is the quotient of $V$ by $U$. It is a vector space itself (briefly, its operations are $(a+U)+(b+U)=(a+b)+U$ and $s(a+U)=(sa)+U$).

More concretely, the affine subspaces associated with $U=\{0\}$ are $a+U=\{a+0\}=\{a\}$ so $V/\{0\}$ is essentially just the points of $V$ itself.

A one dimensional subspace of $\mathbb{R}^n$ is a line through the origin. The corresponding affine subspaces are all lines (not just those through the origin). Specifically, if $U$ is a line through the origin, then $a+U$ is a line parallel to $U$ which passes through $a$.

Likewise, two dimensional subspaces of $\mathbb{R}^n$ are planes through the origin whereas the two dimensional affine subspaces are arbitrary planes.

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  • $\begingroup$ -1; this doesn't answer the question. $\endgroup$ – Qiaochu Yuan Feb 24 '12 at 1:23
  • $\begingroup$ OK, so the "definition" above is not really a good definition, right? It is just a necessary (and not sufficient) condition for a set $B$ to be an affine subspace. $\endgroup$ – Thom Feb 24 '12 at 1:24
  • $\begingroup$ @Thom: it's actually a trivial condition as written. $\endgroup$ – Qiaochu Yuan Feb 24 '12 at 1:25
  • $\begingroup$ Bill, thanks you for your interesting explanation. I choose Qiaochu's answer because it addresses my question more directly, but I think I will look at these concept more deeply, so your answer will be useful to me $\endgroup$ – Thom Feb 24 '12 at 1:30
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    $\begingroup$ While this may not answer the question, it is a very neat and nicely put description of the difference between linear and affine regarding vector spaces $\endgroup$ – Marcos Oct 19 '15 at 2:10

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