A set of vector valued function $y_i(t)$ is linear independent if $$ c_1 y_1(t) + \cdots + c_n y_n(t) = 0 \quad \forall t \in \mathbb{R} \implies c_1 = c_2 = \cdots = c_n = 0. $$
Does this mean that pointwise linear independence of $y_i(t)$ implies linear independence? Since \begin{align} c_1 y_1(t) + \cdots + c_n y_n(t) = 0 \quad \forall t \in \mathbb{R} &\implies c_1 y_1(t_0) + \cdots + c_n y_n(t_0) = 0 \\ &\implies c_1 = c_2 = \cdots = c_n = 0. \end{align}
(I'm assuming $y_i(t)$ are pointwise linear independent at $t_0$.)