Can a variable be linearly independent, but non-linearly dependent? I am reviewing a friend's paper, and they are throwing out variables that are below a certain correlation coefficient value before doing a multiple linear regression model.
Is this a wise thing to do? They are investigating transformations of variables as well, and I feel like a variable that is not linearly correlated could become very informative after a transformation.

Yes, you can have zero correlation when you have a nonlinear dependency.
For example, $y=x^2$ will have a correlation coefficient of zero, and the linear regression will fit a horizontal line. but regression to $y=ax^2+bx+c$ will show the dependency.

