What must $\alpha \in \mathbb{R}^{n}$ be for $\sum_{k=1}^{n} \alpha_{k}x_{k}y_{k}$ to be an inner product?

Question

I am trying to solve the following problem. "Let $x, y \in \mathbb{R}^{n}$. Under which conditions on $\alpha \in \mathbb{R}^{n}$ does the function $f(x,y) = \sum_{k=1}^{n} \alpha_{k}x_{k}y_{k}$ define an inner product on $\mathbb{R}^{n}$ ? "

Naturally, $\alpha$ should be such that the standard criteria for the definition of the inner product is satisfied, but is it actually possible to find these conditions for a general $\alpha$? For each given value of $\alpha$ the standard criteria can be tested, but I am struggling how to answer this question for the general case.

Answer 1

Bilinearity of $f$ always holds.

If $f(x,x)=0$, then

$$ 0=\sum_k \alpha_k x_k^2 $$

We'd like to conclude that each $x_k=0$. What should $\alpha_k$ look like for this to hold?