Adding an endomorphism valued one form to a connection gives a connection

Define ${\eta }_s(p)(v)=B_p(v)s(p)$. In local matrices this is a finite sum of products of smooth coefficients, hence a smooth Hom section. It is real-linear in $s$ and satisfies ${\eta }_{fs}=f{\eta }_s$. Therefore $(\mathrm{\nabla }+B)(fs)=df\otimes s+f\mathrm{\nabla }s+f{\eta }_s=df\otimes s+f(\mathrm{\nabla }+B)s$. This proves the connection axioms.

Adding the zero form fixes $\mathrm{\nabla }$, and adding $B$ then $C$ equals adding $B+C$ by pointwise evaluation. The difference theorem shows any other connection equals $\mathrm{\nabla }+B$ for exactly one $B$, proving transitivity and freeness. For a rank-zero bundle the modeling vector space is zero and the connection space a singleton; an empty base has the same interpretation. In rank one the action is addition of scalar one-forms. No claim of nonemptiness without a given connection, or choice of a preferred origin, enters this affine-space assertion.