The simplest transformations just stretch the axes. A diagonal matrix \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix} scales the x-direction by s_x and the y-direction by s_y. With both factors equal it's a uniform zoom; with them different it squashes one way and stretches the other.

A negative factor flips that axis — a reflection. So \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} mirrors across the y-axis (it sends x \to -x), and \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} mirrors across the x-axis.

Stretch and flip

Set the two scale factors. Positive values stretch or shrink; push one below zero and that axis reflects, turning the grid inside-out across the line. Watch \hat{\imath} and \hat{\jmath} — they always sit on the axes, just lengthened, shortened or reversed.

The telltale signs

Scaling and reflection are the matrices you can read at a glance: everything lives on the diagonal. Their effect on area is just s_x \cdot s_y (each axis contributes its factor), and a single negative factor secretly flips the plane's "handedness" — two ideas the determinant will soon make precise.