What is Affine subspaces; affine maps?

Example 17.9 (Classifying an affine map, start to finish)

Let $f(x, y) = (2x - 1,\ 3y - 4)$ on $\R^2$. Its linear part is $\varphi = \operatorname{diag}(2, 3)$, whose spectrum $\{2, 3\}$ avoids $1$: by the fixed-point criterion proved below (Proposition 17.17), $f$ has exactly one fixed point, found by solving

Recentering at $\Omega$ (set $x = 1 + u$, $y = 2 + v$):

in the frame at $\Omega$, $f$ is its linear part, an anisotropic dilation stretching by $2$ horizontally and $3$ vertically from the center $(1, 2)$. The general lesson: an affine map is “linear map plus location data”, and the location data collapses to one well-chosen origin whenever $1$ is not an eigenvalue. Conversely, translating the origin badly creates the constant terms: affine geometry is the art of choosing where to put $0$.

Example 17.2 (An affine space with no natural origin)

The solution plane $\mathcal E = \{(x, y, z) \in \R^3 : x + y + z = 1\}$ is not a vector subspace ($0 \notin \mathcal E$), but it is an affine space directed by $E = \{x + y + z = 0\}$: for $A, B \in \mathcal E$ the difference $\vect{AB} = B - A$ lands in $E$ (the sums cancel), Chasles is inherited from $\R^3$, and $B \mapsto \vect{AB}$ is bijective onto $E$. No point of $\mathcal E$ is distinguished — any choice of “origin” $O \in \mathcal E$ works equally well, and all the identifications $M \mapsto \vect{OM}$ differ by translations. This is the typical situation: solution sets of inhomogeneous linear problems (linear systems, linear differential equations in Chapter 16) are affine, never linear, and the slogan “particular solution plus kernel” is exactly the statement $\mathcal F = A + F$ of the next definition.

Example 17.18 (Plane isometries, completed)

An affine isometry of the Euclidean plane has linear part in $O(2)$: a rotation $R_\theta$ or a reflection (Year 1 volume). If $\theta \neq 0$: $1 \notin \operatorname{Sp} R_\theta$, so the map is a rotation about a unique center (Proposition 17.17). If the linear part is a reflection: either a reflection in an axis (fixed points exist) or a glide reflection (reflection composed with a translation along the axis, no fixed point). With translations, this is the complete classification of plane isometries.