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I have been working on the following problem.

Find four vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ that point in different directions but are of same lenght, such that every $x \in \mathbb{R}^2$ can be written as $$ x = \sum_{i=1}^4 \langle x, v_i \rangle v_i $$ where $\langle . , . \rangle$ denoted the standard dot-product in $\mathbb{R}^2$.

Can anyone provide useful hint about how to approach ?

My attempt: I figured out that no two vectors among $v_1,v_2,v_3,v_4$ can be orthogonal, otherwise the rest would be zero. After that I am unable to find which vectors hold the above property.

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Choose any orthonormal basis $(e_1,e_2)$ for $\mathbb{R}^2$ and let

$$ v_1 = \frac{e_1}{\sqrt{2}}, v_2 = -\frac{e_1}{\sqrt{2}}, v_3 = \frac{e_2}{\sqrt{2}}, v_4 = -\frac{e_2}{\sqrt{2}}. $$

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