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This question seems embarrassingly basic. Let $X$ be some manifold and $E,E'$ two vector bundles over $X$ with rank $n$ and $n'$ respectively. Now assume to possible refinement we have $E,E'$ defined on the some coordinate atlast $\{U_{i}\}$ with transition maps $$g_{ij}:U_{i}\cap U_{j}\rightarrow GL(n,\mathbb{R});g'_{ij}:U_{i}\cap U_{j}\rightarrow GL(n,\mathbb{R})$$

My question is how to understand the transition map for $E\otimes E'$? I would have guessed it ias $g_{ij}\otimes g'_{ij}$(as matrices), which written down as Kronecker product. But how to prove this rigorously?

For example, suppose $E_{x}$ has bases $\{v_{i}\}$, $E'_{x}$ has bases $\{u_{i}\}$, then $E_{x}\otimes E'_{x}$ has bases $\{v_{i}\otimes u_{i}\}$. The only we can have $g_{ij}\otimes g'_{ij}$ acts on it must be by components: $g_{ij}\otimes g'_{ij}(v_{i}\otimes u_{i})=(g_{ij}v_{i})\otimes (g'_{ij}u_{i})$. But this only showed the existence of such a transition function, not it must be the one we expected. So I venture to ask. I encountered it when Taubes assert that the transitional function for $\wedge^{n}E$ is $\wedge^{n}g_{ij}=\det(g_{ij})$. So I guessed his reasoning in the above.

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