Let $\langle x, y \rangle_{A_k} :=x^{t}A_{k}y$ for all $x,y \in \mathbb{R}^{3}$ with $ A_{k} = \left( \begin{array}{rrr} 1 & k & 0 \\ k & 3 & -2 \\ 0 & -2 & 5 \end{array}\right),k \in \mathbb{R}^{3}.$
For which $k \in \mathbb{R}$ is the vector space $(\mathbb{R}^3, \langle , \rangle_{A_k})$ a euclidean space?
I wasn't really sure what they were trying to have me test here... By a euclidean vector space all I understand is a pair $(V, \langle \cdot, \cdot \rangle)$ made up of a real vector space and a scalar product $\langle \cdot, \cdot \rangle$ on $V$.
So does this mean that the task at hand is to examine $\langle x, y \rangle_{A_k}$ and see for which $k$ it will actually be a scalar product? This entails showing (i) bilinearity, (ii) symmetry, and (iii) positive definiteness ?
I haven't encountered scalar products with a matrix before, is it safe to say that a scalar product of the form $x^{t}Ay$ is symmetric $\Leftrightarrow A$ is symmetric? If so, then (ii) is already taken care of...
I don't really know what should be done for (i)...
For (iii) I think there is some theorem... What are the minors along the diagonal called? I think: all minors along the diagonal have positive determinants $\Rightarrow A$ is positive definite... Is the converse also always true? What about with eigenvalues? All eigenvalues are positive $\Leftrightarrow A$ is positive definite, right?
In this case I tried saying: $\det(1) > 0 \forall k \in \mathbb{R}$, $\det \left( \begin{array}{rr} 1 &k \\ k & 3 \end{array} \right) > 0 \Leftrightarrow k < \sqrt{3}$, and $\det \left( \begin{array}{rrr} 1 & k & 0 \\ k &3 &-2 \\ 0 &-2 &5 \end{array} \right) > 0 \Leftrightarrow k< \sqrt{\frac{11}{5}}$.
Since $\sqrt{\frac{11}{5}} < \sqrt{3}$ we take the stricter condition and say for $\{k \in \mathbb{R} | k< \sqrt{\frac{11}{5}} \}$ the vector space $(\mathbb{R}^{3}, \langle , \rangle_{A_k})$ is a euclidean space. Is this correct?
I also have a question about the way I am wording this: should I say "the vector space" when talking about the same pair with respect to other $k$. I mean assuming that what I claim about $k$ is correct, if $k \geq \sqrt{\frac{11}{5}}$ how would one correctly refer to this: $(\mathbb{R}^{3}, \langle , \rangle_{A_k})$ ? It would definitely not be true to say that $\langle , \rangle_{A_k}$ is a scalar product at least(I assume), but what sort of vector space would that pair be if it weren't euclidean?
As always, any help is very much appreciated :)