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Let $A$ be a $n\times n$ positive definite real symmetric matrix.

I have shown that $\langle x, y\rangle=y^tAx$ satisfies the properties of an inner product.

I want to show that there is a unique $y_0\in \mathbb{R}^n$ such that for each $x\in \mathbb{R}^n$ with $x^t=(x_1, \ldots , x_n)$ we have $x_1+\ldots +x_n=y_0^tAx$.

How can we show that?

Couls you give me a hint?

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    $\begingroup$ Hint : $A$ is invertible. $\endgroup$ – Arnaud D. Sep 3 '18 at 16:37
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I think that has something to do with Riesz representation theorem.

For linear functional $f:\mathbb{R}^n\to \mathbb{R}$ there is a vector $u_0$ such that $f(v) = \langle v,u_0 \rangle$.

In particual $f(v) = x_1+x_2+...+x_n$ vhere $v=(x_1,...,x_n)$.

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  • $\begingroup$ I haven't really understood how the uniqueness implies. Could you explain this further to me? $\endgroup$ – Mary Star Sep 3 '18 at 16:36
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    $\begingroup$ math.stackexchange.com/questions/1738045/… $\endgroup$ – Aqua Sep 3 '18 at 16:37
  • $\begingroup$ Ok! Thank you!! :-) $\endgroup$ – Mary Star Sep 3 '18 at 17:32

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