Linear functional prove

[Chris283]

Let V be a vector space with finite dimensional. Let subset S of V be:

. . . . .$\displaystyle .S^0\, =\, \left\{\,f\, \in\, V^{*}\, \bigl|\bigr.\,\right.$ $\displaystyle f(u)\, =\, 0\,$ $\displaystyle \left. \forall\, u\, \in\, S\,\right\}$

Prove that if W₁ and W₂ are subspaces of V then:

. . . . .$\displaystyle \left(W_1\, \cap\, W_2\right)^0\, =\, W_1^0\, +\, W_2^0$

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I succeed in proving that the right side is contained in the left side of the equation, but I didn't succeed to prove the opposite direction.

I think I should do here something with bases for each of the spaces here, but I don't know how.

Please help me here...