Linear mapping/Matrix/Standard spaces/Example

A linear mapping

$\varphi \colon K^{n}\longrightarrow K^{m}$

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

${}{\begin{pmatrix}y_{1}\\\vdots \\y_{m}\end{pmatrix}}=M{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\,$

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 ${}e_{j}$.