Base change/Transformation matrix/Definition

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}V}$ denote a ${\displaystyle {}K}$-vector space of dimension ${\displaystyle {}n}$. Let ${\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}$ and ${\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{n}}$ denote two bases of ${\displaystyle {}V}$. Let

${\displaystyle {}v_{j}=\sum _{i=1}^{n}c_{ij}w_{i}\,}$

with coefficients ${\displaystyle {}c_{ij}\in K}$. Then the ${\displaystyle {}n\times n}$-matrix

${\displaystyle {}M_{\mathfrak {w}}^{\mathfrak {v}}=(c_{ij})_{ij}\,}$

is called the transformation matrix of the base change from ${\displaystyle {}{\mathfrak {v}}}$ to ${\displaystyle {}{\mathfrak {w}}}$.