Transvection
= Transvection
{title2=$T=I+vf,\quad f(v)=0$}
For a nonzero <vector> $v$ and nonzero <linear functional> $f$ vanishing on $v$, this invertible map fixes the <hyperplane> $\ker f$ pointwise and has determinant one. Its inverse is $I-vf$, since $(vf)^2=0$. A <symplectic transvection> is obtained by taking $f=cB(-,v)$ for a nondegenerate <alternating bilinear form>.