Kernel bundle of a surjective vector bundle morphism

A fiberwise surjective morphism of constant-rank vector bundles has a kernel vector subbundle. For a morphism from a trivial rank-$N$ bundle, a local full-row-rank matrix $F(x)$ gives the continuous kernel projection $I-F(x{)}^T(F(x)F(x{)}^T{)}^{-1}F(x)$. Changing the local frame in the target leaves this projection unchanged. Thus its kernel defines a continuous map into the appropriate Grassmannian.