TITLE: What are affine spaces for?
QUESTION [27 upvotes]: I'm studying affine spaces but I can't understand what they are for. 
Could you explain them to me? Why are they important, and when are they used? Thanks a lot.

REPLY [5 votes]: Affine spaces are important because the space of solutions of a system of linear equations is an affine space, although it is a vector space if and only if the system is homogeneous.
Let $T:V \to W$ be a linear transformation between vector spaces $V$ and $W$. The preimage of any vector $w \in W$ is an affine subspace of $V$. If $w$ is nonzero, then the preimage does not contain $0$ so is not a vector subspace of $V$. However, the preimage of any $w \in W$ is an affine space modeled on the vector space $\ker T$, the kernel of $T$. 
A special case of this example is that the space of solutions of the matrix equation $Ax = y$ (for fixed $y$) is an affine space modeled on the null space of the matrix $A$.