Linear transformation

A linear transformation (also linear map) $T$ of the plane satisfies $T(\vec u + \vec w) = T(\vec u) + T(\vec w)$ and $T(c\vec u) = c\,T(\vec u)$ for all vectors $\vec u, \vec w$ and every real number $c$. Every such map is multiplication by a 2 × 2 matrix $A$, and the columns of $A$ are the images of the basis vectors: $A\hat\imath$ and $A\hat\jmath$. This page shows maps from the plane to itself.

In the picture the two ink arrows are the columns; drag their tips and the whole grid follows, because every point is reached by the same recipe as before. The blue vector v is walked twice: x steps along î and y along ĵ on the original grid (dashed), and x steps along Aî and y along Aĵ on the new one (solid), ending at Av. Reading the same product by rows — each entry of $A\vec v$ is a row of $A$ dotted with $\vec v$ — gives the same numbers. The origin never moves. A line maps to a line or a point. An invertible transformation preserves parallel lines and equal spacing along a line, while lengths and angles can change.

When $\det A = 0$ but $A \ne 0$, the plane collapses to a line. A line through the origin is sent to the origin: this is the kernel. When $A = 0$, every point is sent to the origin, so the kernel is the whole plane. With Preimage, drag the output w to find where it came from. An invertible matrix gives one point $A^{-1}\vec w$; a rank-one matrix gives a line if w lies on the image, and no preimage otherwise. For the zero matrix, the origin has the whole plane as its preimage; other points have none. Solving $A\vec x = \vec b$ is exactly this search.