require "matrix"
sx,sy = 2.0, 3.0;
unit = 2
scale = Matrix[[sx,0], [0, sy]]
scale*unit # => [2.0, 3.0]: scaled point
theta = Math::PI/2 # 90 degrees
rotate = Matrix[[Math.cos(theta), -Math.sin(theta)],
[Math.sin(theta), Math.cos(theta)]]
rotate*unit # [-1.0, 1.0]: 90 degree rotation
scale * (rotate*unit) # [-2.0, 3.0]