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You see $(Y,X)$, you want a relation ship between $X$ and $Y$. You will assume Linear regression I.e you assume it exists $\beta$ such that $Y=X\beta + \epsilon$ and you want to find $\beta$. Solution: $\hat{\beta}=(X'X)^{-1}X'Y$ and $\epsilon = Y-\hat{Y}=Y-X\hat{\beta}=(I-X(X'X)^{-1}X')Y$ So if you apply to your case : $X\to B$ $\beta \to f$ $Y\to R$ ...

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