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?
Best Answer
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.