# What comes first, a vector base or orthogonality?

Pick any three vectors of a vector space randomly (but linearly independent). Then we assign them coordinates:

$$e_1=[1 0 0]$$ $$e_2=[0 1 0]$$ $$e_3=[0 0 1]$$

Therefore now they are orthonormal because $e_i\cdot e_j = \delta_{ij}$. Then this seems to show that all sets of three linearly independent vectors are an orthonormal base, which is false. However I don't see the mistake. I know it must be something silly.

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What do you mean you assign them coordinates? –  Git Gud Sep 23 '13 at 20:16
It's a way to say, "I choose them as a base of the space". –  Ambesh Sep 23 '13 at 20:18