an affine transformation
An affine transformation is a single move that combines a linear distortion with a slide. Picture taking a sheet of stretchy graph paper, stretching and shearing it however you like as long as you keep the grid lines straight and parallel grid lines parallel, then sliding the whole thing across the table. The combined effect — distort linearly, then translate — is an affine transformation. It is the most general kind of map that always sends straight lines to straight lines and keeps parallel lines parallel.
In coordinates the rule has a clean shape. An affine transformation of the plane sends a point (x, y) to A times the column (x, y) plus a fixed shift vector, where A is an invertible 2-by-2 matrix and the shift handles the translation. Writing it out, (x, y) maps to (a x + b y + e, c x + d y + f), with the matrix [a, b; c, d] required to have nonzero determinant so the map can be undone. The matrix part bends and scales directions; the vector part moves everything bodily without distortion. Because the matrix is allowed to scale unequally and to shear, an affine transformation generally changes lengths and angles, but it always preserves three things: collinearity, parallelism, and the ratio in which a point divides a segment along a line. Two special families sit inside: similarities (the matrix is a scaling times a rotation or reflection) and rigid motions (the matrix is a pure rotation or reflection).
Affine transformations are the working tools of affine geometry and are everywhere in applications: computer graphics use them to rotate, scale, shear, and reposition images in one matrix step; they relate any two coordinate systems built from straight axes. A useful fact for proofs is that an affine map of the plane is completely determined by where it sends any three non-collinear points — fix the images of a triangle's vertices and the entire map is pinned down. The common trap is to assume affine maps preserve angles or shape; only the narrower similarity transformations do that, while a general affine map turns a circle into an ellipse.
The map (x, y) -> (2x, x + y) is affine: its matrix [2, 0; 1, 1] has determinant 2, which is nonzero, so it is invertible. It doubles horizontal spread and shears vertically, turning the unit square into a parallelogram. The origin stays put because there is no translation term, and the three corners (1,0), (0,1), (1,1) move to (2,1), (0,1), (2,2).
An affine map is a matrix part (with nonzero determinant) plus a translation; it preserves lines and parallelism.
Affine transformations preserve parallelism and ratios along a line but not angles or absolute length — only the special sub-family of similarities preserves shape, so a general affine map sends a circle to an ellipse.