Is there a smart way to make a base transformation matrix for one base to another?
Here are my bases:
$$ E_1= \begin{bmatrix}-0.4656 & -0.7461 & 0.4760\\ 0.8073 & -0.1377 & 0.5738 \\ -0.3626 & 0.6515 & 0.6664 \end{bmatrix}$$
$$E_2 = \begin{bmatrix} -0.4770 & -0.7463 & 0.4642\\ 0.8064 & -0.1616 & 0.5689\\ -0.3496 & 0.6457 & 0.6789 \end{bmatrix}$$
Think of $E_1$ as the matrix that transforms the standard basis to the basis $\mathcal B_1$, and similarly for $E_2$. To get from $\mathcal B_1$ to $\mathcal B_2$, go from $\mathcal B_1$ to the standard basis and then from the standard basis to $\mathcal B_2$. So this is the matrix product $E_2E_1^{-1}$.