 Kernel bundle of a constant-rank family of linear maps

Let $A_b:V\to W$ vary smoothly with $b$ and have constant rank $r$. The spaces $\ker A_b$ form a vector bundle of rank $\dim V-r$. On a neighborhood where one $r$ by $r$ minor is invertible, the corresponding coordinates are smooth linear functions of the remaining coordinates, giving a local trivialization.