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In [ 3 , Examples 4.16, 4.13], van der Kallen gave an example that the first row map $$\begin{aligned}{} & {} \textrm{GL}_{d+1}(R)\longrightarrow \textrm{Um}_{d+1}(R)/E_{d+1}(R)\\{} & {} \sigma \longmapsto [e_{1}\sigma ] \end{aligned}$$ is not a group homomorphism. In this article, we give a sufficient condition on the ring R so that the first row map becomes a group homomorphism.
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