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1. Let $\displaystyle <x, x>=3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3}$ be a quadratic form in V=R, where $\displaystyle x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ (in the base $\displaystyle {e_{1},e_{2},e_{3}}$.
Find the inner product corresponding to this quadratic form.

Is this that easy that you have to change '' second'' x-es for y (for example to write $\displaystyle 2x_{2}y_{3}$ instead of $\displaystyle 2x_{2}x_{3}$ at the end), or what I have to do?

2.

3. Originally Posted by Denis99
Let $\displaystyle <x, x>=3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3}$ be a quadratic form in V=R, where $\displaystyle x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ (in the base $\displaystyle {e_{1},e_{2},e_{3}}$.
Find the inner product corresponding to this quadratic form.

Is this that easy that you have to change '' second'' x-es for y (for example to write $\displaystyle 2x_{2}y_{3}$ instead of $\displaystyle 2x_{2}x_{3}$ at the end), or what I have to do?
what definition do you have to work with here? If it were me I'd iterate to the result-- start by encoding this with standard basis vectors, i.e.

$\langle x, x \rangle =3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} = \mathbf x^T A \mathbf x$

where $A$ is real symmetric positive definite. Then take the square root of $A$ and let that be your basis i.e. consider
$A^\frac{1}{2} \mathbf x$
where the $kth$ column of $A^\frac{1}{2}$ is denoted by $e_k$ in your text

Originally Posted by steep
what definition do you have to work with here? If it were me I'd iterate to the result-- start by encoding this with standard basis vectors, i.e.

$\langle x, x \rangle =3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} = \mathbf x^T A \mathbf x$

where $A$ is real symmetric positive definite. Then take the square root of $A$ and let that be your basis i.e. consider
$A^\frac{1}{2} \mathbf x$
where the $kth$ column of $A^\frac{1}{2}$ is denoted by $e_k$ in your text
I have to work with definition like this one from definition of inner space in here

(in part Definition)

5. Originally Posted by Denis99
I have to work with definition like this one from definition of inner space in here

(in part Definition)
But I can't figure out how you could understand that and say this

Originally Posted by Denis99
Find the inner product corresponding to this quadratic form.

Is this that easy that you have to change '' second'' x-es for y (for example to write $\displaystyle 2x_{2}y_{3}$ instead of $\displaystyle 2x_{2}x_{3}$ at the end), or what I have to do?
The reality is that one way or another you need to find $A^\frac{1}{2}$