Dot product definition proof

Question

If $w_1$ and $w_2$ are strictly positive, show that the definition $$(x_1,x_2)\dot\ (y_1,y_2)=x_1y_1w_1+x_2y_2w_2,$$ yields an inner product on $\mathbb{R}^2$. Generalize this to $\mathbb{R}^p$.

I am confused on how to do this problem since it doesn't elaborate what $x_1,x_2,y_1,y_2$ are? For these problems should I automatically assume that they are vectors in the Cartesian space? Then I will prove it by letting $x_1=x_i$ and denoting $\vec{x_i}=x_1,...,x_n$ and then proving it by the definition, thus having $w_1(\vec{x_i}+\vec{y_i})=w_1(\langle x_1,...,x_n\rangle\dot\ \langle y_1,...,y_n \rangle)$ ... and preceding conventionally?

Answer 1

You might find this description useful. The standard dot product in $\mathbb R^2$ can be written as a matrix product, with one of the two vectors transposed:

$$\vec{x}\cdot\vec{y} = \left(\begin{array}{cc} x_1 & x_2 \end{array}\right)\left(\begin{array}{c} y_1 \\ y_2 \end{array}\right)$$

The one you are dealing with is another valid inner product which is

$$w(\vec{x},\vec{y}) = \left(\begin{array}{cc} x_1 & x_2 \end{array}\right) \left(\begin{array}{cc} w_1 & 0 \\ 0 & w_2 \end{array}\right) \left(\begin{array}{c} y_1 \\ y_2 \end{array}\right).$$ In general, any expression that looks like this is an inner product when the matrix in the middle is positive definite. The generalization to $\mathbb R ^n$ is

$$\vec{x}\cdot\vec{y} = \left(\begin{array}{cccc} x_1 & x_2 & \cdots & x_n\end{array}\right) \left(\begin{array}{cccc} w_1 & 0 & 0 & 0 \\ 0 & w_2 & 0 & 0 \\ 0& 0& \ddots & \vdots \\ 0&0&\cdots & w_n \end{array}\right) \left(\begin{array}{c} y_1 \\ y_2 \\ \vdots \\y_n \end{array}\right)$$

where the $w_i$ have to satisfy the original conditions.

Answer 2

You are absolutely perfect in assuming the $(x_1,x_2)\;\; and \;\; (y_1,y_2)$ are vectors in $\mathbb R^2$ You can then proceed conventionally - Proving all the properties of the inner product. I am slightly unclear what you intend to do for the second part of the problem.

IMO you have to then take a set $w_i\in \mathbb R : w_i>0 \;\forall\;i=1...p$ and get a generalization of the inner product defined to vectors in $\mathbb R^p$.