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The Linear Transformation Zoo

A 2x2 matrix is a recipe for bending the whole plane, and the recipe is short: it only says where the two basis vectors land. Everything else follows, because the map is linear; grid lines stay straight and evenly spaced, the origin stays put, and parallel lines stay parallel. The two columns of $M$ are exactly the landing spots of $\hat\imath$ and $\hat\jmath$, drawn here as the draggable red and green arrows, so dragging them is the same as typing a matrix. Three numbers read the deformation off at a glance. The determinant is the signed area of the unit square's image, how much the map scales areas, negative when it flips the plane over (a reflection) and zero when it crushes the plane onto a line. The singular values are the longest and shortest stretch, the semi-axes of the ellipse the unit circle becomes. And a real eigenvector is a direction the map leaves pointing the same way, only scaled, drawn as the purple invariant lines; a pure rotation has none, because it turns every direction. Cycle the gallery or drag the vectors and watch all of it move at once.

Figure 1. The linear transformation zoo. Top: the map sends the faint reference grid to the blue grid, the unit square to a parallelogram (area = det), and the unit circle (gold) to an ellipse with its singular-value axes; purple lines are real eigenvector directions, and the red and green arrows are the draggable images of the basis vectors. Bottom: the stretch |M u| against input direction, oscillating between the two singular values. Method: closed-form eigen/SVD of the 2x2 matrix. Source: Strang, Introduction to Linear Algebra, 5th ed., Ch. 6 and Sec. 7.1.

WHAT TO TRY

  • Drag the red and green arrows (the images of the basis vectors): the grid, the parallelogram, the ellipse, and the eigenlines all follow, because the columns are the whole map.
  • Cycle to the reflection: the determinant goes negative as the plane flips over, and the parallelogram's orientation reverses.
  • Cycle to the projection: the determinant is zero, the ellipse collapses to a segment, and one singular value vanishes (the plane is crushed onto a line).
  • Cycle to the rotation: there are no real eigenvectors (every direction turns), and the stretch curve is flat at one (lengths are preserved).