Introduction to Matrix Transformations
A matrix transformation is a systematic way of mapping every point in a 2D plane to a new position using matrix multiplication (and optionally vector addition). This unified framework lets us describe rotations, reflections, stretches, enlargements, and translations with a single algebraic structure.
Matrix Transformation: A matrix transformation in 2D maps a position vector to a new position according to:
where the matrix encodes the linear part of the transformation (rotation, reflection, scaling) and the vector encodes translation.
The power of this approach is that any combination of these geometric operations can be expressed compactly in matrix form, and compositions of transformations reduce to matrix multiplication.
When , the transformation is purely linear , it always maps the origin to itself. A translation alone is not a linear transformation in the strict sense; it requires the added vector.
Reflections
A reflection maps every point to its mirror image across a specified line. The key reflection matrices to know are:
| Line of reflection | Transformation matrix |
|---|---|
| -axis | |
| -axis | |
| Line | |
| Line |
Reflecting about the -axis:
The -coordinate changes sign; the -coordinate is unchanged , exactly what we expect geometrically.
To verify a reflection matrix is correct, check that the basis vectors transform correctly. For reflection about : and , these are the columns of . ✓