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Inverse Map

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A linear map $ \alpha:V\to W$ is injective if and only if $ \operatorname{Ker}\alpha = 0$. In this case an inverse map $ \alpha^{-1}:\operatorname{Im}\alpha\to V$ can be defined by

$\displaystyle w\mapsto v,\quad w = \alpha(v)\, . $

This inverse map is also a linear map. In particular we have

$\displaystyle \alpha^{-1}\circ\alpha(v)=v,\quad \alpha\circ\alpha^{-1}(w)=w $

for all $ v\in V$ and $ w\in\operatorname{Im}\alpha$.
(Authors: Burkhardt/Höllig/Hörner)

see also:

[Annotations]
  automatically generated 2/ 9/2005