This is a question that comes from An Introduction to Statistical Learning.

Consider the fitted values that result from estimating a linear regression without an intercept. That is, suppose you fit the following model: $Y_{i} = 0 + \hat{\beta} X_{i} + \epsilon_{i}$.

You could move this into a summation form, using our definitions of the regression estimators, which would look like the following:

$$ \begin{align} \hat{\beta} = \frac{\sum_{i=1}^{n} x_{i}y_{i}}{\sum_{i'=1}^{n} x_{i'}^{2}} \end{align} $$

(In this notation, we're just using $i$ and $i'$ to distinguish the two summations, but they are both summing over the same data points.)

Show that we can write the fitted values from this regression as: $$\hat{y} = \sum_{i'=1}^{n} a_{i'}y_{i'}$$

In this format, what is $a_{i'}$?

What you should note is that fitted values from a linear regression are linear combinations of the response variables.