We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We denote by $G$ the set of complex polynomials of degree 1.
Verify that $G$ is a group for the law $\circ$.

Determine a pair $(A, \vec{b})$ in $\mathrm{SO}(2) \times \mathbb{R}^2$ such that $M(A, \vec{b}) = I_3$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that $\mathcal{S}$ is a vector space over $\mathbb{R}$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Justify that $T'$ is a distribution on $\mathcal{D}$.

What can be said about the set $\mathcal{M}$ equipped with the operation $*$?

What can be said about the set $\mathcal{M}$ equipped with the operation $*$?