Intersection of affine subspaces is affine

Question

If I have two affine subspaces, each is a translation (or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is also affine, particularly in $\mathbb{R}^d$. My intuition suggests that the resulting space is just a coset of the intersection of the two linear subspaces, but I'm having some trouble arguing this precisely.

Answer 1

Hint: Let $A_1 = x_1 + U_1$, $A_2 = x_2 + U_2$ your two affine subspaces, if $A_1 \cap A_2 = \emptyset$, we are done, otherwise there is an $x \in A_1 \cap A_2$. But then $A_1 = x+ U_1$ and $A_2 = x+ U_2$ ... does this help?

Answer 2

As an exercise you should be able to verify the following.

Theorem. Let $V$ be a linear space, $U \subseteq V$ be a linear subspace, and $x, \, y \, \in V$ be any two elements. Then the following are equivalent

• $x - y \in U$
• $x + U = y + U$
• $(x + U) \cap (y + U) \neq \emptyset$

Now, suppose that $A_1 = x_1 + U_1$ and $A_2 = x_2 + U_2$ are two affine subspaces. Assume $A_1 \cap A_2$ is not empty and choose any $y \in A_1 \cap A_2$. $y \in A_1$ implies that $y = x_1 + u_1$ for some $u_1 \in U_1$, which leads to $y - x_1 \in U_1$. Using the above theorem, this leaves us with $A_1 = y + U_1$. Similarly, you can show that $A_2 = y + U_2$. Again, suppose $z \in A_1 \cap A_2$. This implies that $z = y + u_1 = y + u_2$. This in turn requires that $u_1 = u_2 = u \in U_1 \cap U_2$. Consequently, $z = y + u$ for some $u \in U_1 \cap U_2$. Thus $z \in y + (U_1 \cap U_2)$, implying that $A_1 \cap A_2 \subseteq y + (U_1 \cap U_2)$. Proving the inclusion in the other direction is straight forward and yields the final result as

$$A_1 \cap A_2 = (x_1 + U_1) \cap (x_2 + U_2) = (y + U_1) \cap (y + U_2) = y + (U_1 \cap U_2)$$

for some $y \in (x_1 + U_1) \cap (x_2 + U_2)$.

Answer 3

First step:

If $A_1 = a + U_1$ and $A_2 = b + U_2$ and the intersection is non-empty, then $\exists x \in A_1 \cap A_2\implies \exists u_1\in U_1.u_2\in U_2.x = a + u_1 = b + u_1$. This means we can rewrite $A_1$ and $A_2$ as $$ A_1 = a+U_1 = (x-u_1)+U_1 = x + U_1 $$ and $$ A_2 = b + U_2 = (x - u_2) + U_2 = x + U_2. $$ Second step:

The rest doesn't require any Algebra. We just follow the definition of the intersection of sets and get $$ A_1 \cap A_2 = (x + U_1) \cap (x + U_2) = x + (U_1 \cap U_2).\square $$ Notice that $U_1\cap U_2$ is a vector space because $U_1,U_2$ are.