Linear Transformations

What to observe

1. Drag the tip of î (indigo) or ĵ (teal). The whole grid follows: a linear map is nothing more than where the two basis vectors land.
2. Leave Keep the original grid on and the dotted grid stays where it was, so the solid one has something to move against. Every solid line is still straight, still evenly spaced, still parallel to its neighbours: that, and the origin staying put, is the whole of what linear means.
3. Drag the ring marked v, an ordinary vector that is not a basis vector. The two dashed pieces show how its image is assembled: gox times where î landed, then y times where ĵ landed. Nothing else is ever needed, which is what makes the map linear.
4. Watch the shaded unit square. Its signed area is the determinant; pick Collapse and the plane flattens onto a line asdet → 0 (and the map becomes non-invertible).
5. Turn on Eigenvectors. Those dashed lines are the directions that only stretch, never rotate, scaled by their eigenvalue. Rotations have none (the values go complex).
6. Slide Interpolate or hit Animate to sweep continuously from the identity to the matrix, the moving-picture view of a transformation.