# Show that $W$ is naturally isomorphic to $V / W^0$

If $W$ is a subspace of a finite-dimensional space $V$ , show that $W^*$ is naturally isomorphic to $V^* / W^0$.

Where $W^0$ is the annihilator of $W$ and $V^*$ and $W^*$ are the dual spaces of $V$ and $W$ respectively.

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–  1015 Feb 5 '13 at 2:46

Define

$$\phi:V^*\to W^*\,\,,\,\,\phi(f):=\left.f\right|_W$$

Check that

(1) $\,\,\forall\,\,f\in V^*\,\,,\,\,\phi(f)\in W^*\,$

(2) $\,\phi\,$ is linear

(3) $\,\operatorname{Im}(\phi)=W^*\,$

(4) $\,\,\ker\phi=W^0\,$

(5) Finally, apply the first isomorphism theore (for modules, groups or whatever: it goes well here).

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