Let $E \to M$ and $F \to M$ be smooth vector bundles of ranks $r$ and $s$, respectively, trivialized over the same open cover $(U_i)_{i \in I}$, with transition functions $g^E_{ij}: U_i \cap U_j \to GL(r)$ and $g^F_{ij}: U_i \cap U_j \to GL(s)$. Then the standard vector-bundle operations have transition functions with the following domains and codomains. For the direct sum,

\begin{align*} g^{E\oplus F}_{ij}: U_i \cap U_j \to GL(r+s), \qquad g^{E\oplus F}_{ij}=g^E_{ij}\oplus g^F_{ij}. \end{align*}

For the [tensor product](/page/Tensor%20Product),

\begin{align*} g^{E\otimes F}_{ij}: U_i \cap U_j \to GL(\mathbb{R}^r \otimes \mathbb{R}^s), \qquad g^{E\otimes F}_{ij}=g^E_{ij}\otimes g^F_{ij}. \end{align*}

For the dual bundle,

\begin{align*} g^{E^*}_{ij}: U_i \cap U_j \to GL((\mathbb{R}^r)^*), \qquad g^{E^*}_{ij}=(g^E_{ij})^{-\top}. \end{align*}

For the exterior power, for each $0 \leq k \leq r$,

\begin{align*} g^{\Lambda^k E}_{ij}: U_i \cap U_j \to GL(\Lambda^k \mathbb{R}^r), \qquad g^{\Lambda^k E}_{ij}=\Lambda^k g^E_{ij}. \end{align*}

For the Hom bundle,

\begin{align*} g^{\operatorname{Hom}(E,F)}_{ij}: U_i \cap U_j \to GL(\operatorname{Hom}(\mathbb{R}^r,\mathbb{R}^s)), \qquad g^{\operatorname{Hom}(E,F)}_{ij}(A)=g^F_{ij}A(g^E_{ij})^{-1} \end{align*}

for $A \in \operatorname{Hom}(\mathbb{R}^r,\mathbb{R}^s)$. These functions satisfy the cocycle identities and reconstruct the corresponding operation bundles.