Linear mapping/Matrix/Standard spaces/Example

A linear mapping

$\phi :K^n⟶K^m$

is usually described by the matrix $M$ with respect to the standard bases on the left and on the right. The result of the matrix multiplication

$\left(\begin{array}{c}{y}_1\\ ⋮\\ y_m\end{array}\right)=M\left(\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right)$

can be interpreted directly as a point in $K^m$. The $j$-th column of $M$ is the image of the $j$-th standard vector $e_j$.