Linear mapping/Matrix to basis and reverse/Definition

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}V}$ be an ${\displaystyle {}n}$-dimensional vector space with a basis ${\displaystyle {}{\mathfrak {v}}=v_{1},\ldots ,v_{n}}$, and let ${\displaystyle {}W}$ be an ${\displaystyle {}m}$-dimensional vector space with a basis ${\displaystyle {}{\mathfrak {w}}=w_{1},\ldots ,w_{m}}$.

For a linear mapping

${\displaystyle \varphi \colon V\longrightarrow W,}$

the matrix

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

where ${\displaystyle {}a_{ij}}$ is the ${\displaystyle {}i}$-th coordinate of ${\displaystyle {}\varphi (v_{j})}$ with respect to the basis ${\displaystyle {}{\mathfrak {w}}}$, is called the describing matrix for ${\displaystyle {}\varphi }$ with respect to the bases.

For a matrix ${\displaystyle {}M=(a_{ij})_{ij}\in \operatorname {Mat} _{m\times n}(K)}$, the linear mapping ${\displaystyle {}\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)}$ determined by

${\displaystyle v_{j}\longmapsto \sum _{i=1}^{m}a_{ij}w_{i}}$

in the sense of fact, is called the linear mapping determined by the matrix ${\displaystyle {}M}$.