I need a clarification about the following definition provided by Tu's Introduction to manifolds in section 12.3
A surjective smooth map $\pi : E \to M$ of manifolds is said to be locally trivial of rank $r$ if
(i) each fiber $\pi^{-1}(p)$ has the structure of a vector space of dimension $r$
(ii) for each $p \in M$ there are an open neighborhood $U$ of $p$ and a fiber preserving diffeomorphism $\phi : \pi^{-1}(U) \to U \times \mathbb{R}^r$ such that for every $q \in U$ the restriction $$ \left. \phi \right|_{\pi^{-1}(q)} : \pi^{-1}(q) \to \left\{ q \right\} \times \mathbb{R}^r $$ is a vector space isomorphism. Such an open set $U$ is called a trivializing open set for $E$ and $\phi$ is called a trivialization of $E$ over $U$.
Isn't (ii) essentially saying that for every $p \in M$ the fiber preserving map is represented by a linear transformation? And every linear transformation is continuous (in finite dimension) and therefore we can smoothly change from a vector space $\pi^{-1}(p)$ and $\pi^{-1}(q)$ if $q \in U$ where $U$ is some appropriate neighborhood of $p$?
Is this what the definition of "locally trivial" is trying to cover?