Matrices and transformations

Transformation matrices (about the origin)

• 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)
• Rotation 90° anticlockwise: (0 -1 / 1 0)
• Rotation 180°: (-1 0 / 0 -1)
• Enlargement scale factor k: (k 0 / 0 k)

(Matrices are written row by row; "/" separates the rows.)

Using a matrix

Multiply the matrix by each point written as a column (x / y).

Example: Reflect (2, 3) in the x-axis → (2, -3).

Determinant and area

For (a b / c d), det = ad - bc.

Area of image = |det| × area of object.

Finding a matrix

Use the images of the unit points (1, 0) and (0, 1): they form the columns of the matrix.

Successive transformations

"A followed by B" is the matrix BA (apply A first).

Shear and stretch

• Shear with the x-axis invariant: (1 k / 0 1).
• Stretch parallel to the x-axis, factor k: (k 0 / 0 1).