AHL 3.9 (HL)—Matrix transformations and fractals

A 2×2 matrix maps a point vector to its image: x′=Ax. The columns show where the basis vectors go, so the matrix encodes the geometric action.

Apply the matrix to the column vector in the stated order. Determinant magnitude gives area scale; a negative determinant reverses orientation. To recover an original point, solve with A⁻¹ when it exists.

The matrix [[0,−1],[1,0]] sends (1,0) to (0,1) and (0,1) to (−1,0): a 90° anticlockwise rotation. Multiplying by the row vector would describe a different operation.

Do not read rows as images of basis vectors without checking the convention. A zero determinant means the transformation has no inverse.

The syllabus also includes affine maps $\begin{pmatrix}x'\y'\end{pmatrix}=A\begin{pmatrix}x\y\end{pmatrix}+\begin{pmatrix}e\f\end{pmatrix}$, so translation requires the added vector and cannot be represented by a $2\times2$ matrix alone. In a composition, the rightmost transformation acts first; $BAx$ means apply $A$, then $B$. Iterating one or more transformation rules from an initial shape or point can generate a fractal, while $|\det A|$ remains the area scale for the linear part.