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Prove that the inner product associated with a positive definite quadratic form $q(x)$ is given by the polarization formula $\langle x, y\rangle = \frac{1}{2}[q(x+y) - q(x) - q(y)]$.

How will I be able to do this problem. I know in order to be a positive definite you need the following axioms to be verified: bilinearity, positivity and symmetric. Thus we need that $\langle x, y\rangle = x^TKy$ for $x,y \in R^n$ but how do I go on to apply that here?

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Don't need to apply that, as you wrote, the mentioned axioms are to be verified. How is a pos.def.quadratic form defined? That's what you can use.

But first of all, observe/verify that for given inner product $\langle,\rangle$, with $q(x):=\langle x,x\rangle$ the polarization formula holds.

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    So, instead of verifying that holds I should instead verify that hold?2012-10-07
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    What do you mean by '$\langle x,y\rangle$ holds'? What do we know about $q$?2012-10-07
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    I am a bit confused can you elaborate more on what I need to know and what the question is asking please?2012-10-07
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    I don't know what we can start out from. How is a quadratic form defined in the context? By the matrix $K$? or what?2012-10-07
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    So, have you verified the '*But first of all..*' part? That unfolding $\langle x+y,x+y\rangle$ helps to get back $\langle x,y\rangle$..2012-10-07
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    That is what I am having trouble with since K is not defined in this context. Idk what the question is asking for. I assume it is not defined by the matrix K since it is not given.2012-10-07
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    @diimension: Where is this question from?2012-10-07
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    @wj32 Its practice problems I am doing in my book. The name of the book is "applied linear algebra".2012-10-07
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    @Berci I don't understand what you mean in unfolding.2012-10-07