Let $t_0 \in \mathbb{R}$ is fixed $\mathcal{E}\bigl(\mathbb{R} \times \mathbb{R}^n \bigr)$ be the space of smooth real-valued mappings $f(t,x)$ for $(t,x) \in \mathbb{R} \times \mathbb{R}^n$.
Now, consider the two linear mappings $T_1, T_2 : \mathcal{E}\bigl(\mathbb{R} \times \mathbb{R}^n \bigr) \to \mathcal{E}\bigl(\mathbb{R}^n \bigr)$ defined as \begin{equation} T_1(f):=f(t_0,\cdot) \text{ and } T_2(f):= \bigl(\partial_t f \bigr)(t_0, \cdot) \end{equation}
Then, I wonder if $T_1, T_2$ are linearly independent in the sense that if $a,b \in \mathbb{R}$ satisfy \begin{equation} aT_1(f)+bT_2(f)=0 \text{ for all } f \in \mathcal{E}\bigl(\mathbb{R} \times \mathbb{R}^n \bigr) \end{equation} then, $a=b=0$.
I think this is plausible, but cannot figure out how to prove it rigorously.. Could anyone help me?