I know from the answers at the back of the book that the following map:
$$(S(t)f)(x) = e^tf (x + t),$$ where $S: \mathbb{R} \rightarrow S(V)$ and $V$ is the subspace of all polynomials with real coefficients and $t \in \mathbb{R}, f \in V.$
Is a linear representation (but I do not know how?). Even though I found that $S(t)$ is linear when I put instead of $f$, $\alpha f + \beta f$. I know that for the given map $S$ to be a linear representation it must be a homomorphism, but I see that it is not a homomorphism because $f(x + t +t^`) \neq f(x +t^`) + f(t)$ ?
Could anyone clarify this discrepancy for me please?
EDIT: $(S(t + t^`)f)(x) = e^(t +t`) f (x + t + t`) = e^t e^t` f(x + t + t`)$, then what shall I do next?