Linear/Affine Maps

Given two vector spaces $V$ and $W$, a linear map is a map $T:V\rightarrow W$ such that, for any two scalars $\alpha$ and $\beta$ and two vectors $u, v\in V$, $$T(\alpha\cdot u + \beta\cdot v) = \alpha\cdot T(u) + \beta\cdot T(v)$$

You might also see the condition equivalently expressed as two simpler conditions: $$T(\alpha\cdot u) = \alpha\cdot T(u)$$ $$T(u+v) = T(u)+T(v)$$ It is easy to verify that these two sets of conditions are completely equivalent. As a side note, the linear transformation $T(v)$ is often also denoted $Tv$ when it is clear that suppressing the parentheses will not create confusion.

Given two affine spaces $V$ and $W$, an affine map is a map $F:V\rightarrow W$ such that, for every vector $v\in V$, $F(v) = Tv+b$, where $T$ is some linear transformation and $b\in W$ is a constant vector.

Quite simply, an affine map is the composition of a transformation and a linear map. The only difference is that, in an affine map, the zero vector can be mapped to an arbitrary vector, rather than having to be mapped to the zero vector.