Definition CBM Change-of-Basis Matrix

Suppose that $V$ is a vector space, and $\ltdefn{I_V}{V}{V}$ is the identity linear transformation on $V$. Let $B=\set{\vectorlist{v}{n}}$ and $C$ be two bases of $V$. Then the change-of-basis matrix from $B$ to $C$ is the matrix representation of $I_V$ relative to $B$ and $C$, \begin{align*} \cbm{B}{C}&=\matrixrep{I_V}{B}{C}\\ &=\matrixrepcolumns{I_V}{C}{v}{n}\\ &=\left\lbrack \left.\vectrep{C}{\vect{v}_1}\right| \left.\vectrep{C}{\vect{v}_2}\right| \left.\vectrep{C}{\vect{v}_3}\right| \ldots \left|\vectrep{C}{\vect{v}_n}\right. \right\rbrack \end{align*}