If $f(x)=ax+b$ is linear, why is $f(\gamma x) \neq \gamma f(x)$

Question

The definition of a linear function is that it satisfies (among other things): $$ f(\gamma x) = \gamma f(x) $$

Then, why does for $f(x)=ax+b$, the following inequality appear: $$ f(\gamma x) = a\gamma x+b \neq \gamma(a x+b) = \gamma f(x) $$

Which means that:
$$ f(\gamma x) - \gamma f(x) = a\gamma x+b - \gamma(a x+b) = a\gamma x+b - a\gamma x - \gamma b) = b(1-\gamma) \neq 0 $$

What am I missing here?

EDIT TIL that there's a difference between a linear function and a linear map. Clarified in more detailed here for the curios: intutive difference between linear map/transformation vs linear function

score 3 · Accepted Answer · answered Sep 26 '19 at 18:08

3

Note that

$$f(x)=ax+b$$

is said to be "linear" because its graph is a straight line but (for $b\neq 0$) it is not linear in the sense of "linear map" (e.g. $f(x)=ax$).

answered Sep 26 '19 at 18:08

user

154,566

So rotation and scaling is allowed for linear maps, but not translations? – Void Sep 26 '19 at 18:24
1

@Void Yes, translations are not described by linear map (it is an affine map). – user Sep 26 '19 at 18:26

score 1 · Answer 2 · answered Sep 26 '19 at 18:05

1

You are missing the fact that, unless $b=0$, the map $x\mapsto ax+b$ is not a linear map.

answered Sep 26 '19 at 18:05

José Carlos Santos

427,504

So it's a linear function but not a linear map? Could you elaborate on the distinction please? – Void Sep 26 '19 at 18:08
1

It's a linear function since its graph is a stright line, but that doesn't turn it into a linear map. – José Carlos Santos Sep 26 '19 at 18:10
So a linear map is always a linear function but the other way around is not always a given? – Void Sep 26 '19 at 18:13
1

That is correct. – José Carlos Santos Sep 26 '19 at 18:15
1

I always called $ax+b$ an affine function (in France), I'm surprised it's called linear in some other countries, therefore the confusion. – zwim Sep 26 '19 at 19:33

score 0 · Answer 3 · answered Sep 26 '19 at 18:05

0

You're missing the second half of the definition, namely $f(x+y)=f(x)+f(y)$. Note that $f(x)=ax+b$ fails this condition in general, and is therefore not linear by this definition.

answered Sep 26 '19 at 18:05

79037662

1,379

If $f(x)=ax+b$ is linear, why is $f(\gamma x) \neq \gamma f(x)$

3 Answers3