What is affine transformation? Which distribution families are closed under affine transformation?
Affine transformation(left multiply a matrix), also called linear transformation(for more intuition please refer to this blog: A Geometrical Understanding of Matrices), is parallel preserving, and it only stretches, reflects, rotates(for example diagonal matrix or orthogonal matrix) or shears(matrix with off-diagonal elements) a vector(the same applies to many vectors/a matrix), and the "non-affine"(also a type of projective transformation as explained in the comment by @whuber) transformation may be like the first example in the following diagram:
More generally speaking affine transformation has the following three properties:
straight lines preserved
parallel lines preserved
ratios of lengths along lines preserved (midpoints preserved)
So I look here: http://mathworld.wolfram.com/AffineTransformation.html
It is a rotation. All points on a line, stay on the same line.
Per @Luca: It can have scaling, shear, translation as well. No bending. Straight lines are always straight.