📂Matrix Algebra

Affine Transformation

Definition

Simple Definition

Suppose a matrix $A$ and a vector $\mathbf{b}$ are given. The transformation that multiplies the matrix $A$ to the vector $\mathbf{x}$ and adds $\mathbf{b}$ is called an affine transformation. $$ \mathbf{x} \mapsto A \mathbf{x} + \mathbf{b} $$

Complex Definition ¹

If $f : V \to V$ defined on a vector space $V$ satisfies the following condition for any scalar $\lambda$, then $f$ is called an affine transformation. $$ f \left( \lambda x + \left[ 1 - \lambda \right] y \right) = \lambda f (x) + \left[ 1 - \lambda \right] f (y) $$

Explanation

In contexts such as machine learning or geometry, multiplying by matrix $A$ refers to taking a linear transformation like a rotation transformation, and adding vector $\mathbf{b}$ refers to a translation.

In fields familiar with mathematics, the expression affine transformation is often used essentially to mean just multiplying by a matrix, because an affine transformation like $\mathbf{y} = A \mathbf{x} + \mathbf{b}$ can be thought of in a block matrix form, where it’s possible to adjust by adding $1$ to the bottom of the two vectors $\mathbf{y}$ and $\mathbf{x}$ as follows. $$ \begin{bmatrix} \mathbf{y} \\ 1 \end{bmatrix} = \begin{bmatrix} A & \mathbf{b} \\ \mathbf{0}^{T} & 1 \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ 1 \end{bmatrix} = \begin{bmatrix} A \mathbf{x} + \mathbf{b} \\ 1 \end{bmatrix} $$