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Prove that the inner product associated with a positive definite quadratic form q(x) is given by the polarization formula $⟨x,y⟩=1/2[q(x+y)−q(x)−q(y)]$.

How will I be able to do this problem?

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Since the association between quadratic form $q$ and innner product $\langle \cdot, \cdot \rangle$ is given by $q(x) = \langle x, x \rangle$, this is more-or-less direct calculation using the linearity of the inner product. In particular, note that

$$q(x+y) = \langle x+y, x+y\rangle = \langle x, x\rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle = q(x) + 2 \langle x,y\rangle + q(y),$$ using linearity to expand $\langle x+y,x+y\rangle$ and symmetry to transform $\langle y,x\rangle$ to $\langle x, y\rangle$.

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Thank you very much! I understand how you used symmetry and linearity to expand the function but how will this particular function relate to the polarization formula.? –  diimension Oct 9 '12 at 7:24
    
Um... subtract $q(x) + q(y)$ and divide by 2. This gives you the polarization formula. –  Johannes Kloos Oct 9 '12 at 9:03
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