Transformations (AQA GCSE Further Maths): Revision Notes
📚Revision Notes
Matrix transformations
What are transformations?
When we talk about transformations in mathematics, we're describing how points move or change position. If a point P with coordinates (x, y) gets moved to a new position P' with coordinates (x', y'), we say that point P has been mapped to its image point P'.
Note
Think of it like taking a photograph - the original point is the object, and the image point is what appears in the photo after some change has been applied.
Understanding matrix transformations
A transformation can be completely described using a special type of matrix called a transformation matrix. This matrix tells us exactly how every point in the plane will be moved.
Here's how it works: when we want to understand what a transformation matrix does, we look at where it sends two special points:
- The point (1, 0)
- The point (0, 1)
If our transformation sends (1, 0) to the point (a, c) and sends (0, 1) to the point (b, d), then our transformation matrix looks like this:
Important
This matrix contains all the information we need to transform any point in the coordinate plane.
Position vectors and how they work
When we use matrices to transform points, we need to write the coordinates in a special way called a position vector. A position vector shows a point's coordinates in a vertical arrangement, with the x-coordinate written above the y-coordinate.
For example, if we have a point (-1, 5), its position vector would be written as:
Note
The position vector represents the point's location measured from the origin (0, 0), and this vertical format is essential for matrix multiplication to work properly.
How matrix transformations work
To find where a point goes after a transformation, we multiply the transformation matrix by the point's position vector. The mathematical process looks like this:
If we have a transformation matrix:
And we want to transform a point (x, y), we calculate:
Important
The result (x', y') tells us the coordinates of the image point after the transformation has been applied.
Worked example - step by step
Example
Worked Example: Finding Image Points
Problem: Find the image of point (-2, 5) after applying the transformation defined by the matrix:
Solution:
Step 1: Write the point (-2, 5) as a position vector:
Step 2: Multiply the transformation matrix by the position vector:
Step 3: Calculate each component of the result:
- First row:
- Second row:
Step 4: Write the result as a position vector:
Step 5: Convert back to coordinate form: The image point is (11, 9).
Common types of transformations
Different transformation matrices produce different types of geometric changes:
- Identity transformation: The matrix with 1s on the main diagonal and 0s elsewhere leaves all points unchanged
- Reflexion transformations: Certain matrices can reflect points across lines like the x-axis or y-axis
- Rotation transformations: Other matrices rotate points around the origin by specific angles
Note
Remember that rotations are assumed to be anticlockwise unless specifically stated otherwise.
Key Points to Remember
Summary
Key Points to Remember:
- A transformation matrix completely describes how every point in the plane will be moved
- Position vectors are written vertically with the x-coordinate on top
- To find an image point, multiply the transformation matrix by the position vector
- The result of matrix multiplication gives you the coordinates of the transformed point
- Always check your arithmetic carefully when multiplying matrices - small errors can lead to wrong answers