Matrix transformations

A matrix with columns u and v sends (1,0) to u and (0,1) to v. Transform the four corners of a unit square to identify the image. Rotations and reflections act about the origin unless a different construction is explicitly supplied.

For successive transformations A then B, a column vector becomes BAv. The rightmost matrix acts first. Translations cannot be represented by an ordinary 2 × 2 linear transformation matrix.

Worked example

Apply [[0,−1],[1,0]] to (2,3).

1. Use the column vector [[2],[3]].
2. The image is [[−3],[2]].
3. The unit vectors rotate 90° anticlockwise.

Answer: (−3,2), a 90° anticlockwise rotation about the origin

Revise first: Matrix multiplication.