Chapter 9 : Linear transformation

1. An aside on analysis

A function from a set $X$ to a set $Y$ is a rule that tells how elements of both sets are associated to each other.

The element $y \in Y$ associated, under the function, to the element $x \in X$ is the so-called image.
The element $x \in X$ associated, under the function, to the element $y \in Y$ is the so-called pre-image.

Example I

The function represented in Figure 9.1 associates the grades of the students belonging to the class D1 in the last statistics examination. We have :

• $X= \{\text{Alex}, \text{Marie}, \text{Yoko}, \text{Ryu}, \text{Seiko}, \text{Katy}\}$. $Y=\{1,2,3,4,5,6\}$.
• The image of $\text{Marie}$ under the function is $4$.
• The pre-image of $5$ under the function is $\{\text{Yoko}, \text{Katy}\}$.
• $\text{Alex}$ does not have any image (absent from the examination).
• $1$ and $2$ both do not have any pre-image.

Figure 9.1 : function

2. Definition

A linear mapping or linear transformation is a function satisfying the two following conditions:

In other words, for a linear transformation the order things are processed does not matter. You can either first perform the arithmetic operation (multiplication or addition) and second apply the result to the function, or reversely. Both lead to the same result.

Example II

Let $j$ :

$$\begin{equation} j \colon \, \mathbb{R} \to \mathbb{R} \tag{3} \\ \quad x \mapsto 3x \end{equation}$$

$(3)$ is a linear transformation since $(1)$ and $(2)$ both are satisfied :

$$\eqalignno{ j(a + b) &= 3(a+b) = \underbrace{3a}_{j(a)} + \underbrace{3b}_{j(b)} &(4) \cr j(\lambda a) &= 3(\lambda a) = \lambda \underbrace{3a}_{j(a)} & (5)\cr }$$

Example III

Let $k$ :

$$\begin{equation} k \colon \, \mathbb{R} \to \mathbb{R} \tag{6} \\ \quad x \mapsto -2x +9 \end{equation}$$

$(6)$ is not a linear transformation since neither $(1)$ nor $(2)$ are satisfied :

$$\eqalignno{ k(a + b) &= -2(a+b)+9 = -2a -2b+9 \ne \underbrace{-2a+9}_{k(a)} + \underbrace{-2b+9}_{k(b)} &(7) \cr k(\lambda a) &= -2(\lambda a)+9 \ne \lambda ( \underbrace{-2a+9}_{k(a)} ) & (8) \cr }$$

3. Matrix and linear transformation

Based on the properties of the matrices operations, it can be shown that :

$$\begin{equation} A(\vec{a}+ \vec{b}) = A\vec{a}+ A\vec{b} \tag{9} \end{equation}$$

$$\begin{equation} A(\lambda \vec{a}) = \lambda A\vec{a} \tag{10} \end{equation}$$

A matrix multiplying a vector satisfies the two conditions $(1)$ and $(2)$ and is thus a linear transformation.

4. Visualization

Linear transformations are for example used in video games for rotating objects.

Example IV

Let $l$ :

$$\begin{equation} l \colon \, \mathbb{R}^2 \to \mathbb{R}^2 \\ \left( \begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \right) \mapsto \left( \begin{smallmatrix} 3 & 0 \\ 0 & 2 \end{smallmatrix} \right) \left( \begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \right) \tag{11} \end{equation}$$

Figure 9.2 illustrates in dashed lines the linear transformation of a shape. The transformation is in that case a stretching (also called scaling).

Figure 9.2 : linear transformation

Figure 9.3 : customized linear transformation (credits Disney-Pixar)

Let’s define a linear transformation of your own and observe the result !

Wrap-up