How to incorporate a non-linear relationship in a multiple regression model? Suppose the beta coefficients of a weighted multiple regression model is given by the matrix formulation:
${\boldsymbol \beta} = ({\bf X}^T{\bf W}{\bf X})^{-1}{\bf X}^T{\bf Wy}$
Suppose that the dependent variable was correlated to one of the independent variables exponentially by its square (eg. $v_d=v_i^2$). Is it possible to incorporate such a non-linear relationship into the model using this formulation, or does it require a whole different method?
 A: If this is the correct form, you could fit a linear regression with $E(Y|X) = b_0 + b_1 X_1 + b_2 X_2$ where $X_2$ is $X_1 \times X_1$.
This would present a relationship where $Y$ is quadratic in $X_1$ but linear in the beta parameters. 
If you're not sure if a quadratic term is appropriate, and if you don't care much about interpreting the beta estimates easily, you could fit a restricted cubic spline of $X_1$ and model 
$E(Y|X) = b_0 + \text{rcs}(X1).$ 
This is more flexible and requires less assumptions that may create out of sample stability. 
Edited: I wrote "instability" above, but I believe RCS actually is more stable out of sample for predictions when the rcs is formulated in a reasonable manner. So, I changed it to "stability".
The overall short answer to your question: yes, because you can call $Z$ the new independent variable which is just a quadratic (or other) transformation of some $X_j$, then plug $Z$ into your beta calculation. The key would be linearity in the beta parameters.  
