Let $\mathcal{B} = \{v_1, \ldots, v_m\}$ be a basis for the domain $\mathbb{R}^m$ and $\mathcal{C} = \{u_1, \ldots, u_n\}$ be a basis for the codomain $\mathbb{R}^n$.

The matrix $\mathbf{V}$ has columns $v_1, \ldots, v_m$ (domain basis), $\mathbf{U}$ has columns $u_1, \ldots, u_n$ (codomain basis), and $\mathbf{A}$ is the standard matrix of $T$.

Matrix Representation:

$ \mathbf{B} = \mathbf{U}^{-1} \mathbf{A} \mathbf{V} $

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

$ [T(\mathbf{x})]_{\mathcal{C}} = \mathbf{B} \, [\mathbf{x}]_{\mathcal{B}} $

Key Ideas:

• $\mathbf{B}$ represents the transformation in non-standard bases
• When both bases are standard, $\mathbf{B} = \mathbf{A}$ (the standard matrix)
• This is a similarity transformation when domain = codomain