Express a function in mathematical notation

Question

I am familiar in reading mathematical notation but I have troubles when I try to use it. I have the next question. What I want is to express in math notation, all the linear functions that goes from $R^2$ to R, how could i do that? The option that i thought is:

$f:R^2 \to R$ $\quad$ where $\quad f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)$ | $ x,y\in R^2 \land \alpha,\beta\in R$

Is this correct or may be am I omitting or doing something wrong?

Answer 1

You could first give your child a name. For instance define $$ L(\mathbb R^2,\mathbb R):=\{f:\mathbb R^2 \to \mathbb R\vert\,f\text{ is linear}\}. $$

Recall that a map $f:\mathbb R^2 \to \mathbb R$ is called linear if $$ f(v+\lambda\cdot w)=f(v)+\lambda\cdot f(w) \forall v,w\in\mathbb R^2,\lambda\in\mathbb R. $$

You could also write $$ L(\mathbb R^2,\mathbb R)=\{f:\mathbb R^2 \to \mathbb R,\, (x,y)\mapsto \alpha x+\beta y\,\vert\,\alpha,\beta \in\mathbb R\} $$ However this is something that requires a proof starting from the definition of $L(\mathbb R^2,\mathbb R)$.

Answer 2

I think that $\operatorname{Hom}(\Bbb R^2,\Bbb R)$ is quite a standard notation for your set: $$\operatorname{Hom}(\Bbb R^2,\Bbb R):=\{f\colon\Bbb R^2\to\Bbb R\mid f\text{ is linear}\}$$