# symmetry-translation-point-linearity-failure.mp4

Chapter section: Translations and Function Representation

Caption (verbatim):

Shifting a number line is not a linear operation

Image description (verbatim):

Point translation failing linearity

## Before the animation

Chapter lines 505–511:

```math
T_a(x_1)+T_a(x_2)
=
(x_1+a)+(x_2+a)
=
x_1+x_2+2a.
```

Chapter lines 513–513:

These are not equal unless $a=0$. Translation does act sensibly on a number line, but it is not a linear operation when acting on this space.

## After the animation

Chapter lines 521–521:

On the other hand, the space of functions on a number line does form a linear representation. Instead of translating the point $x$, we translate a function by shifting its argument:

Chapter lines 523–525:

```math
(T_a f)(x)=f(x-a).
```