Take a continuous-time dynamical system $\Sigma=(\mathbb{T},\mathbb{W},\mathfrak{B})$ with $\mathbb{T}=\mathbb{R}$, $\mathbb{W}=\mathbb{R}$ and all sinusoidal signals with period $2\pi$.
i.e. $w:\mathbb{R}\to\mathbb{R}\in\mathfrak{B}\iff\exists A\in\mathbb{R}_{+}$ and $\varphi\in[0,2\pi)$ such that $w(t)=A\sin{(t+\varphi)}$.
I want to work out whether this dynamical system is linear or not. We know that $\Sigma$ is linear if $\mathbb{W}$ is a vector space and $\mathfrak{B}$ is a linear subspace of $\mathbb{W}^{\mathbb{T}}$.
$\mathbb{W}=\mathbb{R}$, which is a vector space.
$\mathbb{W}^{\mathbb{T}}=\mathbb{R}^{\mathbb{R}}=\{\text{all maps } \mathbb{R}\to\mathbb{R}\}$
$\mathfrak{B}=\{w:\mathbb{R}\to\mathbb{R}|\exists A\in\mathbb{R}_{+}\text{ and }\varphi\in[0,2\pi)\text{ such that }w(t)=A\sin{(t+\varphi)}\}$
$w$ maps $\mathbb{R}$ to $\mathbb{R}$, which should imply that $\mathfrak{B}\subset\mathbb{W}^{\mathbb{T}}$, but since $\sin{(t+\varphi)}\ne\sin{(t)}+\sin{(\varphi)}$ and thus is not linear, does that imply that the dynamical system is not linear?