A [[Matrix]] that [[Linear Transformation|Transforms]] a [[Vector]] in some [[Basis]] $\mathcal B=\set{\vec b_{1}, \vec b_{2},\dots, \vec b_{n}}$ to another [[Basis]] (ex. the [[Standard Basis]] $\mathcal S$). $\huge \mathcal{C}_{\mathcal B \to \mathcal S} = \mat{ [\vec b_{1}]_{\mathcal B} & \cdots & [\vec b_{n}]_{\mathcal B} } $ If $\mathcal{C}_{\mathcal B\to \mathcal S}$ is [[Inverse Function|Invertible]], then the [[Change of Basis Matrix]] from $\mathcal S\to\mathcal B$ is the [[Logical Inverse|Inverse]] of the [[Matrix]] that maps $\mathcal B \to \mathcal S$. $\huge \mathcal{C}_{\mathcal S \to \mathcal B} = {\mathcal{C}_{\mathcal B \to \mathcal S} }^{-1} $ For a [[Change of Basis Matrix]] from $\mathcal B_{1}\to \mathcal B_{2}$ can be expressed as: $\huge \mathcal{C}_{\mathcal{B_{1}\to B_{2}}}= \mathcal{C_{S\to B_{2}}}\circ\mathcal{C_{B_{1}\to S}} $