Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 9 / c / i

Here both regressor columns are already centered, so $Z=XV$ and $X=Z{V}^T$. Writing $\beta =({\beta }_1,{\beta }_2{)}^T$, the linear predictor is $X\beta =Z{V}^T\beta$. ThusThe intercept stays ${\beta }_0$, because centering made the transformation a rotation of the regressor columns without an added translation.