Amoebas and coamoebas of linear spaces

We give a complete description of amoebas and coamoebas of $k$-dimensional very affine linear spaces in $({\mathbb{C}}^{\ast }{)}^n$. This include an upper bound of their dimension, and we show that if a $k$-dimensional very affine linear space in $({\mathbb{C}}^{\ast }{)}^n$ is generic, then the dimension of its (co)amoeba is equal to $\min \{2k,n\}$. Moreover, we prove that the volume of its coamoeba is equal to ${\pi }^{2k}$. In addition, if the space is generic and real, then the volume of its amoeba is equal to $\frac{{\pi }^{2k}}{2^k}$.