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I am posed with the following question.

Let $\left(V:F\right)$ be a two dimensional vector space. $\beta = \{x_1, x_2\}$ is a basis of $V$ and $\beta^* = \{\phi_1, \phi_2\}$ is the dual basis of $V$. If $ \beta^{'} = \{x_1+2x_2\, 3x_1+4x_2\} $ is also a basis of $V$, find the dual basis of this in terms of $\phi_1$ and $\phi_2$.

What would you suggest for this question?

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    Are you sure you didn't mean "the dual basis for $V^*$"? The dual basis lives in the dual space, unless you have an inner product you're not telling us about.2011-03-28
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    I'm having a hard time following you, I'm just trying to learn this subject and I got this question from a friend. The answer's also written there, ${\beta'}^{\ast} = \{-2\phi_1+\phi_2,\frac{-3}{2}\phi_1+\frac{1}{2}\phi_2\}$2011-03-28
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    Which part gives you trouble? I'd be glad to elaborate. Also, your friend's answer is incorrect.2011-03-28
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    see http://math.stackexchange.com/questions/29314/basis-and-linear-functionals/29329#293292011-03-28

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