I am not sure if the title correctly captures my question(I apologize if it doesn't).
In linear regression whose matrix form is $$Y = Xb + E$$ where $X=[X_1,X_2]$.
If the error of using only $X_1$ (i.e $ Y = X_1b+E$ ) is $x_1$, and the error of using $X_2$ (i.e $ Y=X_2b+E$) is $e_2$, what is the range the error using both columns, i.e $Y=[X_1,X_2]b + E$?
I tried to project $Y$ to $X_1$, $X_2$, and the $span\{X\}$ using Gram-Schmidt, but I think I might overcomplicate the problem and haven't got anything yet.