basis_transform.py

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

b1 = np.array([
    [1, 0, 0],
    [0, 1, 0],
    [0, 0, 1]
])

t = 15 / 180 * np.pi

b2 = np.array([
    [np.cos(t), -np.sin(t), 0],
    [np.sin(t), np.cos(t), 0],
    [0, 0, 1]
])

v1 = np.array([1, 1, 0]) / 1.414
v2 = np.array([np.sqrt(3)/2, 0.5, 0]).T  # v1 in basis 2

v2 = np.linalg.inv(b1) @ b2 @ v2  # transform back to v1

ax.quiver3D(0, 0, 0, *b1, color='k')
ax.quiver3D(0, 0, 0, *v1, color='k', alpha=0.5)
ax.quiver3D(0, 0, 0, *b2, color='tomato')
ax.quiver3D(0, 0, 0, *v2, color='tomato', alpha=0.5)

ax.set_xlim(-0.2, 1)
ax.set_ylim(-0.2, 1)
ax.set_zlim(0, 1)
ax.set_xlabel("X")
ax.set_ylabel("Y")
ax.set_zlabel("Z")
plt.show()