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I know that we can defined different dot product on a vector space. Yet when we are working in an orthogonal basis we have :

$$\langle x,y\rangle = x_1y_1 + \cdots +x_ny_n$$

So does it mean that every dot product is actually the same (because it’s just the product of the coordinates) ?

I know this must be false, but I don’t understand why because the fact that $\langle x,y\rangle = 0$ when $x,y$ are orthogonal means that every dot product is actually the product of the coordinates of the two vectors.

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  • $\begingroup$ If $\langle x,y \rangle$ is an inner product so is $a\langle x,y \rangle$ for any $a>0$. $\endgroup$ – Kabo Murphy Sep 18 '18 at 23:25
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"Orthogonal" is always defined with respect to a particular inner product. Different inner products on the same finite-dimensional vector space will have different orthonormal bases. Of course you can map one orthonormal basis to another, producing an isomorphism of inner product spaces.

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