Real Affine Sets and Subspaces
Why
What are the affine sets in terms of subspaces?
Affine sets which are subspaces
The subspaces of $\R ^n$ are the affine sets which contain the origin.
$M \subset \R ^n$ is a subspace if and only
if $M$ is affine and $0 \in M$.
($\Rightarrow$)
Suppose $M$ is a subspace.
Then $0 \in M$.
Also $\alpha x + \beta y \in M$ for all
$\alpha , \beta \in \R $ and $x, y \in \R ^n$
In particular, $(1- \lambda )x + \lambda y \in
M$ for all $\lambda \in \R $, $x, y \in
\R ^n$.
In other words, $M$ contains the line through
$x$ and $y$.
($\Leftarrow$) Suppose $M$ is affine and $0
\in M$.
$M$ is closed under scalar multiplication since
\[ \alpha x = (1 - \alpha )0 + \alpha x \]
is in the line through $0$ and $x$. $M$ is closed under vector addition since\[ (1/2)(x + y) = (1-1/2)x + (1/2)y \]
is in the line through $x$ and $y$. Thus, $x+y = 2(1/2)(x+y) \in M$.Affine sets as translated subspaces
Suppose $M \neq \varnothing$ is affine.
Then there exists a unique subspace $L$ and
vector $a \in \R ^n$ for which $M = L + a$.
Moreover,
\[ L = M - M = \Set{x - y}{x,y \in M}. \]
First, uniqueness.
Suppose $L_1$ and $L_2$ are subspaces parallel
to $M$.
We will show that $L_2 \supset L_1$ (and
similarly, $L_1 \supset L_2$).
Since $L_1$ and $L_2$ are both parallel to
$M$, they are also parallel to each other.
Consequently, there exists $a \in \R ^n$ with
$L_2 = L_1 + a$.
Since $0 \in L_2$ (it is a subspace, after
all), $-a \in L_1$.
Since $L_1$ is a subspace, $a \in L_1$.
So $x + a \in L_1$ for every $a \in L_1$,
and so $L_2 = L_1 + a \subset L_1$.
A similar argument gives $L_1 \supset L_2$.
If $y \in M$, then $M + (-y) = M - y$ is
a translate of $M$ containing zero (since $y -
y = 0$).
In other words, the affine set $M - y$ is a
subspace.
This, then, is the unique subspace parallel to
$M$.
Since $y$ was arbitrary, the subspace parallel
to $M$ is $L = \cup_{y \in M} M - y =
\Set{x - y}{x,y \in M}$.