gen-define-affine-transformations-and-give-one-short-worked-exam-0510

Define affine transformations and give one short worked example.

# Affine Transformations ## Definition An **affine transformation** is a function between vector spaces that preserves points, straight lines, and planes. It consists of a linear transformation followed by a translation. In general form: $$T(\mathbf{x}) = A\mathbf{x} + \mathbf{b}$$ where: - $A$ is a matrix (the linear part) - $\mathbf{b}$ is a vector (the translation part) - $\mathbf{x}$ is the input vector **Key properties preserved:** - Collinearity (points on a line stay on a line) - Ratios of distances along a line - Parallelism ## Worked Example Transform the point $(2, 3)$ using the affine transformation: $$T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix}$$ **Solution:** $$T\begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix}$$ $$= \begin{pmatrix} 1(2) + 2(3) \\ 0(2) + 1(3) \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix}$$ $$= \begin{pmatrix} 8 \\ 3 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 7 \\ 7 \end{pmatrix}$$ The point $(2, 3)$ is mapped to $(7, 7)$.