# When is it true that F(x + y) = F(x) + F(y)

#### Peter_CSR

##### New member
Hallo,

I am new here and my apology in advance for my not so good math English... I have never used it before....

So, as the title states, for what kind of functions/mapping(...? Display? Projection? I am virtually unable to find correct English translation for X: A->B, defined on any set, such as real numbers, vectors, matrixed... I hope you get my drift...sorry!) is the following statement true: F(x + y) = F(x) + F(y)

I think this works only and only for linear mapping(?) and none other.

Please, note my question refers to precise math definition and I would like to ask for answers only university students/staff if you don't mind. I hope it is all right like that.

Thank you very much!

#### tkhunny

##### Moderator
Staff member
Hallo,

I am new here and my apology in advance for my not so good math English... I have never used it before....

So, as the title states, for what kind of functions/mapping(...? Display? Projection? I am virtually unable to find correct English translation for X: A->B, defined on any set, such as real numbers, vectors, matrixed... I hope you get my drift...sorry!) is the following statement true: F(x + y) = F(x) + F(y)

I think this works only and only for linear mapping(?) and none other.

Please, note my question refers to precise math definition and I would like to ask for answers only university students/staff if you don't mind. I hope it is all right like that.

Thank you very much!
This condition is one of the two requirements to have a "Linear Map". The other being scalar multiplication. Can you think of an example of a mapping WITH Additivity but WITHOUT Scalar Multiplication?

Since it is NOT the case that $$\displaystyle Additivity(Alone)\implies Linear\;Map$$, I would like to see your proof of the assertion that $$\displaystyle Not((Linear\;Map))\implies Not(Additivity)$$.