Lin Independent Sections iff Trivial Bundle This is from Hatcher: Let $(E,B,p)$ is a vector bundle.

If one has $n$-linearly independent sections, the map $h:B \times \mathbb{R}^n \to E$ given by $h(b,t_1,···,t_n)= \sum_i t_i s_i(b)$ is a linear isomorphism in each fiber, and is continuous since its composition with a local trivialization $p^{-1}(U) \to U \times \mathbb{R}^n$ is continuous.

I've had no trouble up til now. This part just seems vague to me. How is the composition continuous?
 A: With regards to your specific query, the local trivialisations commute with the fibrewise linear structure on $E$. So if $\varphi:p^{-1}(U)\xrightarrow{\cong}U\times \mathbb{R}^n$ is such a local trivialisation then for each $b\in U$ we have
$$\varphi\left(\sum_{i=1}^n t_i\cdot s_i(b)\right)=\sum_{i=1}^n t_i\cdot \varphi (s_i(b))$$
where the right-hand side is interpreted as the sum in $\mathbb{R}^n\cong\{b\}\times\mathbb{R}^n$. The point is that the sections $s_i$ are continuous with respect to the variable $b\in U$ so if for each $i=1,\dots,n$ we let $\hat s_i:U\rightarrow \mathbb{R}^n$ be the composite
$$\hat s_i:U\xrightarrow{s_i} E_U\xrightarrow{\varphi}U\times \mathbb{R}^n\xrightarrow{pr_2} \mathbb{R}^n$$
then we get a family of continuous $\mathbb{R}^n$-valued maps on $U$, and our first equation tells us that the composite $\varphi\circ h|_{U\times\mathbb{R}^n}:U\times\mathbb{R}^n\rightarrow U\times \mathbb{R}^n$ is equal to the composite
$$U\times \mathbb{R}^n\xrightarrow{\Delta\times 1}U^{n+1}\times\mathbb{R}^n\xrightarrow{shuf}U\times (\mathbb{R}\times U)^n\xrightarrow{1\times\prod(1\times\hat s_i)}U\times (\mathbb{R}\times \mathbb{R}^n)^n\xrightarrow{1\times m^n}U\times (\mathbb{R}^n)^n\xrightarrow{1\times\oplus}U\times\mathbb{R}^n$$
where $\Delta:U\rightarrow U^{n+1}$ is the $(n+1)$-fold diagonal, the second map $shuf$ shuffles the coordinates appropriately, $m:\mathbb{R}\times\mathbb{R}^n\rightarrow \mathbb{R}^n$ is scalar multiplication and $\oplus:\mathbb{R}^n\times\dots\times\mathbb{R^n}\rightarrow\mathbb{R}^n$ $(x_1,\dots,x_n)\mapsto x_1+\dots+x_n$ is the iterated vector addition in $\mathbb{R}^n$.
It should be clear from this presentation that $\varphi\circ h$ is continuous (and even smooth if you work in the $C^\infty$ category).
