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So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again 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 sone trouble arguing this precisely.

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up vote 3 down vote accepted

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?

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