Specifically referring to where it says "Show that any transvection $\tau$ has the form $x\mapsto x+f(x)u$ where $f(x)$ is a linear function and $u$ is a vector such that $f(u)=0$" is this a typo?
Shouldn't it be that for any $x$ linearly independent to $u$ that $f(x)=0,$ and $f(u)\neq0?$
Or am I just missing something here?