Foliated vector fields span tangent bundle

Question

Suppose we have a foliation on a manifold $M$ which will be call $F$. Foliated vector fields are those $X$ for which: $$[X,T] \in TF$$ for all $T \in TF$. It is easy to see that locally if we have a foliated chart than: $$X=\sum_{i=1}^p a_i(x,y) \frac{\partial}{\partial x_i}+ \sum_{i=1}^q b_i(y) \frac{\partial}{\partial y_i}$$ where as usually $TF = span \{ \frac{\partial}{\partial x_i} \}$. My question is wheather or not foliated vector fields span $TM$ locally -in that foliated chart?

Answer 1

Yes they span locally. You may take ${\partial \over{\partial x_i}}$ and ${\partial \over{\partial y_j}}$ which are foliated and generate $TM$.