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Polyethylene-based resins have the disadvantage that they are not resistant to low temperatures in use, but a particular form of these resins is very suitable for producing articles which, for example, are to be used for ski boots or ski binding components, for which the resulting polyethylene-based articles must withstand the relatively low temperatures occurring in this use.
In the production of such articles, the use of mixtures of a polyethylene-based resin and a polypropylene-based resin is known. The mixture is normally homogenized in a kneader, where it is cooled and then extruded in a first extruder.
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Expected value of $\langle f_i(x_1), f_i(x_2) \rangle$
Suppose we have an inner product space $(E,\langle\cdot,\cdot\rangle)$ which is finite-dimensional over $\mathbb{R}$. For every $i = 1, \cdots, n$, let $f_i : E \to E$ be a linear map. I have two related questions:
$\text{(i)}$ Is there a way of representing $f_i$ by a matrix whose entries are expected values of $\langle x_j, y_k \rangle$ for some fixed $i$ and $j,k$? That is, does there exist matrices $A_i$ and $B_i$ such that $$f_i(x) = A_i \langle x, x_j \rangle x_k + B_i \langle x, y_j \rangle y_k \quad \text{for every } x \in E?$$
$\text{(ii)}$ Given the statement of the first question, suppose we also have a collection of numbers $c_{i
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