Stop thinking of a matrix as a table, and start thinking of it as an action: a rule that takes in a vector and moves it to a new one, transforming the entire plane at once. A linear transformation is a special, very well-behaved kind of motion. It must keep two promises:

• grid lines stay straight, parallel and evenly spaced, and
• the origin stays fixed.

In symbols, a transformation T is linear if it respects addition and scaling: T(\vec{u}+\vec{v}) = T(\vec{u}) + T(\vec{v}) and T(c\vec{v}) = c\,T(\vec{v}). No bending, no curving, no shifting off the origin — just a clean stretch-and-skew of space.

Watch space deform

Drag the slider to apply the transformation gradually. The grid stretches and shears — yet every line stays a line, parallel lines stay parallel, and the centre never moves. That rigidity is exactly what "linear" buys us, and it's why a single matrix can capture the whole motion.

What stays linear, what doesn't

Rotations, stretches, reflections and shears are all linear. Bending a line into a curve is not; nor is sliding the whole plane sideways (that moves the origin — it's an affine move, not a linear one). Because linear maps preserve the grid, they're completely pinned down by where just two arrows go — the subject of the very next page, and the reason every linear transformation can be written as a matrix.