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?