## 1. Analytical aside

A **function** from a set $X$ to a set $Y$ is rule telling how elements of both sets are associated 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 respect to the last examination of statistics. 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 at the examination).
- $1$ and $2$ both do not have any pre-image.

## 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 dot matter. You can either first proceed the **arithmetic operation** (multiplication or addition) and second **apply** the result to the function, or reversely. Both leads to the same result.

#### Example II

Let $j$ :

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

#### Example III

Let $k$ :

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

## 3. Matrix and linear transformation

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

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

## 4. But more concretely ?

Linear transformations are for example used in image processing or for making objects rotate in video games.

#### Example IV

Let $l$ :

In Figure 9.2 is illustrates in dashed line the linear transformation of a shape. The transformation is in that case a stretching.

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

## Recapitulation

A matrix **multiplying** a vector is a **linear transformation** (also called linear mapping).

This kind of mapping stretches (or shrinks) a vector or rotates a vector or combines both.