vectorization of matrix

The vectorization of a $m\times n$ matrix  $A=(a_{ij})$  the matrix to a column vector $\text{vec}(A)$, which consists of all columns of $A$ stacked in :

$$\text{vec}(A):={\left(a_{11}{a}_{21}\mathrm{\dots }{a}_{m1}{a}_{12}{a}_{22}\mathrm{\dots }{a}_{m2}\mathrm{\dots }\mathrm{\dots }{a}_{1n}{a}_{2n}\mathrm{\dots }{a}_{mn}\right)}^{⊺}$$

The mapping vec from the vector space formed by the $m\times n$ matrices to the vector space of the column vectors of the $mn$ is apparently a linear transformation.