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**arbolis** Let $\displaystyle \alpha _1=(1,2)$ and $\displaystyle \alpha _2 =(3,4)$. Let $\displaystyle \text{( , )}$ be an inner product in $\displaystyle \mathbb{C}^2$ such that :

$\displaystyle (\alpha _1, \alpha _1)=1$

$\displaystyle (\alpha _1, \alpha _2)=1+2i$

$\displaystyle (\alpha _2, \alpha _2)=\frac{1}{2}$

Calculate $\displaystyle (\alpha , \beta) \forall \alpha$ and $\displaystyle \beta \in \mathbb{C}^2$.

My attempt : I tried to figure out how is definied the inner product but I didn't find. After all I'm not even sure I should start the problem this way. If I should, then it's not an obvious thing to do... I need help on this.