The determinant of transvections

A transvection in a $K$-vector space $V$ is a linear map $T(f,v)$ of the form $x\mapsto x + f(x) v$, where $f\in V^*$ is a linear form and $v\in V$ is a vector such that $f(v)=0$. It is known that such a linear map is invertible, with inverse given by $f$ and $-v$. More precisely, one has $T(f,0)=\mathrm{id}$ and $T(f,v+w)=T(f,v)\circ T(f,w)$. In finite dimension, these maps have determinant $1$…