Kernel Of surrjective linear bundle map? $E$ is a vector bundle on $M$. let $\phi:E\longrightarrow TM$ be a surjective linear bundle map. Is $\ker\phi$ vector sub-bundle of $E$?
 A: The answer is yes. 
Let $\phi:E\to F$ be a smooth surjective linear bundle map between two vector bundles of ranks $n+k$ and $k$ (resp.)  on some  manifold $M$.  Then $Ker(\phi)$ has constant rank  $n$,  and the question is only if its locally trivial, ie can be spanned  by $n$ locally defined  smooth sections of $E$, pointwise linearly independent. 
Fix a point $x_0\in M$ and trivialize $E$ and $F$ around $x_0$ by  sections $e_1 ,\ldots,e_{n+k} $, and $f_1 ,\ldots, f_k $ (resp.), such that $e_1 , \ldots, e_n $ span $Ker(\phi)$ at $x_0$. Let $\phi(e_i)=\sum_{j=1}^k a_{ij}f_j$ for $i=1, \ldots, n$, and $\phi(e_{n+i})=\sum_{j=1}^k b_{ij}f_j$ 
for $i=1, \ldots, k.$ Then  $A=(a_{ij})$, $B=(b_{ij})$, are $n\times k$ and $k\times k$ (resp.) matrix-valued functions defined on $M$ around $x_0$, such that $A(x_0)=0$, hence, by the surjectivity of $\phi$,   $B(x_0)$ is  invertible, hence $B(x)$ is also invertible near $x_0$ and $B^{-1}(x)$ is smooth.  
Now let $s_i=e_i-\sum_{j=1}^k c_{ij}e_{n+j}$, $i=1,\ldots, n$, where $C=(c_{ij})=B^{-1}A.$ Then 
you can easily check that $s_1,\ldots, s_n$ are smooth section of $E$ that  span $Ker(\phi)$ near $x_0$. QED. 
