Originally Posted by

**NonCommAlg** i can solve the problem provided that the following is true (so you need to tell me if such a thing basically exists in your textbook or not):

Theorem (i just made it up!): suppose $\displaystyle V$ is a finite dimensional vector space over $\displaystyle F$ and $\displaystyle W$ is a subspace of $\displaystyle V^*.$ then for any linear map $\displaystyle \varphi: W \to F,$ there exists $\displaystyle v \in V$

and a linear map $\displaystyle \tilde{\varphi} : V^* \to F$ such that $\displaystyle \tilde{\varphi} |_W=\varphi$ and $\displaystyle \tilde{\varphi}(g)=g(v),$ for all $\displaystyle g \in V^*.$

the reason that i think there must be such a theorem in linear algebra is that we have the same situation in ring theory, i.e. if $\displaystyle R$ is self-injective and $\displaystyle I,J$ are two ideals

of $\displaystyle R,$ then $\displaystyle Ann(I)+Ann(J)=Ann(I \cap J).$