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I am not quite sure what this question is asking for:

Given $f(\vec{x})=x^2+xy+y^2+yz+z^2+xz$, find a basis for the corresponding inner product on $\mathbb R^3$.

(I was told that there is an obvious orthogonal basis -- obtainable by inspection)

I don't even know what "corresponding inner product" means...

Thanks.

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    A positive-definite quadratic form is essentially an inner product. To get an explicit expression for the symmetric bilinear form, look up the polarisation formula.2011-12-02
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    Thanks, @ZhenLin, but I still don't quite understand this, would you mind elaborating?2011-12-02
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    @ZhenLin: What does it mean to find a basis for an inner product?2011-12-02
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    The problem is a bit vague, but note $f(\vec{x}) = \vec{x}^T A \vec{x}$ where $A = \left[\begin{matrix} 1 & 1/2 & 1/2\\1/2 & 1 & 1/2\\1/2 & 1/2 & 1\end{matrix}\right]$. Given positive semidefinite matrix $B$, one can define an inner product: $\langle \vec{x}, \vec{y} \rangle_B = \vec{x}^T B \vec{y}$. See the connection?2011-12-02
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    I think "find a basis for an inner product" means "find a basis B for the vector space such that B is orthonormal with respect to the inner product."2011-12-02
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    @GerryMyerson: THanks!2011-12-02

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