1.4.4—Matrix transformations

A 2×2 matrix maps a vector (x,y) to a new vector by combining its components. Its columns are the images of the standard basis vectors, so the matrix describes rotation, reflection, stretch, shear or a combination.

Composition is matrix multiplication, and the order records which transformation happens first. Determinant describes area scaling; eigenvectors, when present, keep their direction under the transformation.

The matrix [[0,−1],[1,0]] sends (1,0) to (0,1) and (0,1) to (−1,0), a 90° anticlockwise rotation.

Read the coordinate convention carefully: multiplying on the left by a column-vector matrix is not interchangeable with using row vectors.