# Bases of dual spaces

I am given that dual to the basis $B=\{v_1,v_2,...,v_n\}$ of the vector space $V$ is the dual basis $\{f_1,f_2,...,f_n\}$ of $V^*$ where $V^*$ is the dual space of $V$.

How do I find the dual basis with respect to another basis, $B'$ of $V$?

I know that any basis of $V$ are composed of vectors which are linear combinations of $\{v_1,v_2,...,v_n\}$. And then...?

Thanks.

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Think about what it means to be a dual basis. Consider how this can be converted to solving a system of linear equations. –  Willie Wong Nov 18 '11 at 14:57
@WillieWong: Thanks, I had a brain-dead episode and muddled up the definition of "dual basis"... –  TNT Nov 18 '11 at 15:56

How do you know $\{f_{1},...,f_{n}\}$ ? If you know the relation between $\{v_{1},...,v_{n}\}$ and $\{f_{1},...,f_{n}\}$ then you can find the dual basis with respect to another basis.