Matrix and linear transformation

Idea

A matrix sends every vector in the plane to a new place. Where the basis $\hat{ı},\hat{ȷ}$ goes decides the whole linear transformation.

Interaction

• Free transform: drag the entries $a,b,c,d$, and watch the images of $\hat{ı}$ and $\hat{ȷ}$ pull the grid. The sidebar shows $\det$ and its sign
• Special transform: choose Rotation, Scale, Shear, or Reflection, and move Parameter to see what each one does to the grid
• Composition: adjust A rotation and B shear, and compare the two orders

Suggested order: Free transform, then Special transform, then Composition.

What to notice

• Where the two column vectors are sent, the whole grid follows. The matrix entries are only the coordinates of those two images.
• The determinant reads the area and the orientation this same transform carries away. Flattened to a line or a point, there is no way back.
• Two transforms in a row still act on the same plane. Swapping the order usually misses the same grid, because the picture in the middle has already changed.

Related

• Eigenvectors and stretch factors
• Linear transformation grid
• Affine transformation pattern
• Rotation and scaling, composed
• Iterated affine fractal

Further reading

• Linear map (Wikipedia)
• Determinant (Wikipedia)