Transforming the unit square
- A 2 × 2 matrix transforms points: multiply the matrix by the point written as a column vector.
- The columns of the matrix show where the unit vectors go: the first column is the image of [1; 0] and the second is the image of [0; 1]. Use this to find any transformation matrix.
- The unit square has vertices (0, 0), (1, 0), (1, 1) and (0, 1). Its image shows the effect of the transformation.
Standard matrices
- Reflection in the x-axis: [1 0; 0 −1]. Reflection in the y-axis: [−1 0; 0 1].
- Reflection in y = x: [0 1; 1 0]. Reflection in y = −x: [0 −1; −1 0].
- Rotation 90° anticlockwise about the origin: [0 −1; 1 0]. Rotation 180°: [−1 0; 0 −1]. Rotation 90° clockwise (270° anticlockwise): [0 1; −1 0].
- Enlargement, scale factor k, centre the origin: [k 0; 0 k].
Combining transformations
- Transformation A followed by transformation B is the single matrix BA. The first transformation goes on the right.
- Example: reflection in the x-axis, A = [1 0; 0 −1], followed by rotation 90° anticlockwise, B = [0 −1; 1 0]. BA = [0 1; 1 0], which is a reflection in y = x.
- The image of (3, 2) under [0 −1; 1 0] is (−2, 3).
Key terms
- Transformation matrix
- A matrix that moves points in a particular way.
- Unit square
- The square with vertices (0, 0), (1, 0), (1, 1) and (0, 1).
- Image
- The result of a transformation.
- Unit vector
- A vector of length 1, such as [1; 0] or [0; 1].
- Combined transformation
- One transformation followed by another.