Matrix transformations
Objective. Represent plane transformations and compositions.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
Each 2×2 matrix represents a linear transformation of the plane, mapping the point (x,y) to the point given by multiplying the matrix by the column vector (x,y). Standard transformations include rotations, reflections and enlargements, each with a characteristic matrix form, such as [[cosθ,−sinθ],[sinθ,cosθ]] for a rotation by θ anticlockwise. Combining two transformations corresponds to multiplying their matrices, with the matrix applied second placed on the left. For example, reflecting then rotating a shape is represented by (rotation matrix) × (reflection matrix).
Worked example
A transformation is a rotation of 90° anticlockwise about the origin. Find the image of the point (3, −2) under this matrix transformation.
- 1The rotation matrix for 90° anticlockwise is [[0,−1],[1,0]]
- 2Apply to the vector (3,−2): new x = (0×3) + (−1×−2) = 2
- 3New y = (1×3) + (0×−2) = 3
- 4So the image point is (2, 3)
Answer: (2, 3)
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Practice
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Start practisingCommon mistake. Students multiply transformation matrices in the order the transformations are described rather than the reverse (rightmost acts first). For 'reflect then rotate', write Rotation × Reflection, not Reflection × Rotation.
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