import numpy as np

def change_basis(v, dims):
    """
    :param v: vector of any dimension
    :param dims: basis of new dimension matching dimensions of v
    :return: the same vector v in new coordinate space
    """
    return [np.dot(v, x) / np.dot(x, x) for x in dims]

change_basis([5, -1], [[1, 1], [1, -1]])

change_basis([10, -5], [[3, 4], [4, -3]])

change_basis([2, 2], [[-3, 1], [1, 3]])

change_basis([1, 1, 1], [[2, 1, 0], [1, -2, -1], [-1, 2, -5]])

change_basis([1, 1, 2, 3], [[1, 0, 0, 0], [0, 2, -1, 0], [0, 1, 2, 0], [0, 0, 0, 3]])