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Da $\{u_{1},u'_{2}\}$ eine Basis für $U$ ist, gilt wegen Linearität von $\phi_{1}$ - \begin{mathe}[mc]{rcccccccl} + \begin{mathe}[mc]{rcccccl} \range(\phi_{1}) &= &\phi_{1}(\vectorspacespan\{u_{1},u'_{2}\}) &= &\vectorspacespan\{\phi_{1}(u_{1}),\phi_{1}(u'_{2})\} - &= &\vectorspacespan\{v_{1},v'_{2}\} - &= &V, + &= &\vectorspacespan\{v_{1},v'_{2}\}, \end{mathe} und damit