computational geometry – For two sets of points find if second one is result of linear transformation of the first

Say we have two sets of points in vector-2 space (In actuality need to solve this problem in vector-3 space but decided to start with a simpler problem). The points in the second set are the result of transformation of the points in the first set. It’s not known which point in the first set corresponds to which point in the second set. What needs to be calculated is if the points of the second set can be seen as the linear transformation of the first set. So basically finding whether there exists a 2 by 2 matrix, which if applied to the first set gives the second set. And if so calculating it.

Should a problem like this be solved using computational geometry, like sweep line algorithm to compare all points between sets?