A shear slides the plane sideways by an amount that grows with height — like pushing the top of a deck of cards while the bottom stays put. A horizontal shear leaves \hat{\imath} alone but tips \hat{\jmath} over to the side:

The entry k is how far the top leans. Squares become slanted parallelograms, yet — and this is the surprise — the area never changes. Sliding doesn't add or remove any space, it only skews it.

Lean it over

Slide k. Horizontal lines stay put; everything above the x-axis drifts sideways in proportion to its height, so vertical lines tilt into parallel diagonals. The base vector \hat{\imath} is pinned; only \hat{\jmath} moves.

Area-preserving, and oddly important

Because a shear keeps area, its determinant is 1. Shears might look like a curiosity, but they're the elementary moves behind row operations — every step of solving a linear system is, geometrically, a shear or a scale. Master shears and you've quietly understood how equations get solved.