Linear Transformations from $\mathbb R$ to $\mathbb R$

Question

Is my iff statement correct?

$f:\mathbb R\to\mathbb R$ is a Linear Transformation iff there exists a unique $a\in\mathbb R$ such that for all $x\in\mathbb R$, $f(x)=xf(a)$

So if I am given any linear transformation from $\mathbb R$ to $\mathbb R$ it should be of this form right?

Answer 1

It is better to say thta $f$ is linear iff $f(x) = ax$ for a unique $a \in \Bbb R$ and moreover $a = f(1)$

If $f$ is linear $$f(x) = f(x 1) = x \ f(1)$$ so $a = f(1)$

If $f(x) = ax$ then you can easily check that it is linear and $f(1) = a$ $$f(x + y) = a (x + y) = ax + ay = f(x) + f(y)$$ $$f(bx) = abx = bax = b f(x)$$ for every $x,y,b \in \Bbb R$

Answer 2

If $f\neq0$, $a$ is unique.

If $f\neq0$ we can use the relation

$$ \dim(\ker(f))+\dim(\Im(f))=\dim(\Bbb R)=1. $$

If it were $\dim(\ker(f))=1$, we would have $\dim(\Im(f))=0$, hence $\Im(f)=\{0\}$, so $f=0$, excluded. Therefore $\ker(f)=\{0\}$, i.e. $f$ is injective.

Suppose then $f(x)=xf(a)=xf(b)$ for all $x\in\Bbb R$. Putting $x=1$ (or any $x\neq 0$) we obtain $f(a)=f(b)$ and injectivity gives $a=b$.