Solved questions (see the comment below):
- For a vector bundle $E\rightarrow B$, if there are $n$ linearly independent sections {$s_1,…,s_n$} (i.e. trivial bundle). We take a linearly independent subset of $E_b$ for $b\in B$: {$s_1(b),…,s_n(b)$}. Does it mean for any $b\in B$, $s_1(b)\ne0$, $s_2(b)\ne0$,..., $s_n(b)\ne0$? >>>> True.
- For a vector bundle cannot admit $n$ linearly independent sections (i.e. non-trivial bundle), if I still select a collection of sections {$s_1, …,s_n$}, does it mean there is at least one subset of $E_b$ is zero, i.e. $s_i(b)=0$? >>> Not True.
Updated question:
- A non-orientable rank $n$ vector bundle $E\rightarrow B$, for any collection of sections {$s_1, …,s_n$}, there is at least one subset of $E_b$ is zero, i.e. $s_i(b)=0$, $b\in B$?
Consider $\gamma\oplus\varepsilon^1 \to S^1$ where $\gamma$ and $\varepsilon^1$ denote the non-trivial and trivial line bundles over $S^1$ respectively. Note that $w_1(\gamma\oplus\varepsilon^1) = w_1(\gamma) \neq 0$, so the bundle is non-orientable. As $\varepsilon^1$ is trivial, it admits a nowhere-zero section $s$. Now set $s_k = (0, ks)$ and note that for $k \neq 0$, the section $s_k$ is a nowhere-zero section of $\gamma\oplus\varepsilon^1$. In particular, $\{s_1, s_2\}$ is a collection of nowhere-zero sections of $\gamma\oplus\varepsilon^1$. That is, the answer to your question is no.