Is it valid to apply an operation to coordinates on a graph? Ex: $2(a,b) = (2a, 2b)$?

Question

As the title says, is it valid to do something like $2(a,b)$ where $(a,b)$ are points on a graph, such that $(a,b)$ becomes $(2a,2b)$ ? or is this not valid because coordinates cannot be changed using an operation?

Answer 1

The $xy$-plane is also known as $\Bbb{R}^2$, and it forms a vector space over $\Bbb{R}$. So the operation $\lambda (a,b) = (\lambda a, \lambda b)$ is valid by definition of a vector space. Look up the definition of vector space to see.

Now if $(a,b)$ is a coordinate of a graph of a function $y = f(x), \ f : \Bbb{R} \to \Bbb{R}$, where by graph you mean $G(f) = \{ (a,b) \in \Bbb{R}^2 : b = f(a) \}$, then if $(a,b) \in G(f)$, then $\lambda (a,b)$ is $\notin G(f)$ possibly. I.e. it isn't necessary that $G(f)$ is closed under scalar multiplication in $\Bbb{R}^2$.

Take for example the well-known $f(x) = x^2$. Then $(2, 4) \in G(f)$, but $2(2, 4) \notin G(f)$.

What are necessary and sufficient conditions for $G(f)$ to be closed under scalar multiplication by some $\lambda$? Well,

$$ \forall (a,b) \in G(f), \lambda (a,b) \in G(f) \\ \iff \forall a \in \Bbb{R}, f(\lambda a) = \lambda f(a) $$

Thus the graph of all linear functions is closed under scalar multiplication by $\lambda$ for all $\lambda \in \Bbb{R}$, for example.