The matrix $\mathbf{V}$ has columns $v_1, \ldots, v_n$ (initial basis vectors) and $\mathbf{U}$ has columns $u_1, \ldots, u_n$ (final basis vectors).

Change of Coordinates Matrix:

$ \mathbf{S} = \mathbf{U}^{-1} \mathbf{V} $

If $[\mathbf{x}]_{\mathcal{B}}$ represents coordinates in basis $\mathcal{B}$, then:

$ [\mathbf{x}]_{\mathcal{C}} = \mathbf{S} \, [\mathbf{x}]_{\mathcal{B}} $

Key Ideas:

• $\mathbf{S}$ transforms coordinate vectors from basis $\mathcal{B}$ to basis $\mathcal{C}$
• The inverse $\mathbf{S}^{-1} = \mathbf{V}^{-1}\mathbf{U}$ goes from $\mathcal{C}$ back to $\mathcal{B}$
• If $\mathcal{C}$ is the standard basis, then $\mathbf{S} = \mathbf{V}$ converts to standard coordinates