Relate coefficients under linear feature transformation

Imagine you are fitting a linear regression model and comparing two different feature parameterizations. Untransformed features: x1, x2. Transformed features:

• $z_1 = x_1 + x_2$
• $z_2 = x_1 - x_2$ Model A (untransformed features):
• $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \varepsilon$ Model B (transformed features):
• $y = \gamma_0 + \gamma_1 z_1 + \gamma_2 z_2 + \varepsilon$ Questions:

1. How are $(\beta_1, \beta_2)$ and $(\gamma_1, \gamma_2)$ related?
2. Are Model A and Model B equivalent (that is, do they have the same expressive power / can they produce identical fitted values)? Under what assumptions might they behave differently in practice? Overview: This item tests understanding of linear regression and linear-algebraic feature transformations, focusing on how coefficient vectors relate under reparameterization and how parameter interpretability changes; Category/domain: Statistics & Math, position type: Data Scientist.