With two given bases and vector coordinates in the first basis how do I find coordinates in the second basis?

Answer 1

This is not a geometry question. It is a linear algebra question. Load the vectors of the first basis into the columns of a matrix A. Load the vectors of the second basis into the columns of another matrix B. Invert matrix B, call the inverse C. The change of basis matrix is CA. Multiply the change of basis matrix by the coordinate vector with respect to the first basis to get the coordinate vector with respect to the second basis.