Transformations using a Matrix
What is a transformation matrix?
- A transformation matrix is used to determine the coordinates of an image from the transformation of an object
- reflections, rotations, enlargements and stretches
- Commonly used transformation matrices include
- (In 2D) a multiplication by any 2x2 matrix could be considered a transformation (in the 2D plane)
- This can be done similarly in higher dimensions
- An individual point in the plane can be represented as a position vector,
- Several points, that create a shape say, can be written as a position matrix
- A matrix transformation will be of the form
- where represents any point in the 2D plane
- is a given matrix
How do I find the coordinates of an image under a transformation?
- The coordinates (x’, y’) - the image of the point (x, y) under the transformation with matrix are given by
- Similarly, for a position matrix
- A calculator can be used for matrix multiplication
- If matrices involved are small, it may be as quick to do this manually
- STEP 1
Determine the transformation matrix (T) and the position matrix (P)
The transformation matrix, if uncommon, will be given in the question
The position matrix is determined from the coordinates involved, it is best to have the coordinates in order, to avoid confusion
- STEP 2
Set up and perform the matrix multiplication required to determine the image position matrix, P’
P’ = TP
- STEP 3
Determine the coordinates of the image from the image position matrix, P’
How do I find the coordinates of the original point given the image under a transformation?
- To ‘reverse’ a transformation we would need the inverse transformation matrix
- i.e. T-1
- For a 2x2 matrix the inverse is given by
- where
- A calculator can be used to work out inverse matrices
- You would rearrange
Examiner Tip
- Read the question carefully to determine if you have the points before or after a transformation
Worked example
A quadrilateral, Q, has the four vertices A(2, 5), B(5, 9), C(11, 9) and D(8, 5).
Find the coordinates of the image of Q under the transformation .