I’m not sure what you mean by a change of basis making a nonlinear system linear. A linear system is one where solutions add as elements of a vector space. That’s true no matter what basis you express it in.

It depends on parameterization.

For example, if you prameterize the x,y coordinates of a plane-circular trajectory in terms the angle theta, it's nonlinear function of theta.

However, if you parameterized a point in terms of the tuple (cos \theta, sin \theta) it comes out as a scaled sum. Here we have pushed the nonlinear functions cos and sin inside the basis functions.

A conic section is nonlinear curve (not a line) when considered in the variables of and y. However, in the basis of x^2, xy, y^2, x, y it's linear (well, technically affine).

Consider the Naive Bayes classifier. It looks nonlinear till one parameterized it in log p, then it's linear in log-p and log-odds.

If one is ok with dimensional basis this linearisation idea can be pushed much further. Take a look at this if you are interested

https://math.stackexchange.com/questions/4471490/a-proper-ap...