I am reading Kane's book on Reflections and Reflection Groups and am having difficulties with some basic geometric notions - see attachment Kane paes 6-7.

On page 6 in section 1-1 Reflections and Reflection Groups (see attachment) we read:

" We can define reflections either with respect to hyperplanes or vectors. First of all, given a hyperplane $\displaystyle H \subset E $ through the origin, let L = the line through the origin that is orthogonal to H. So $\displaystyle E = H \oplus L $"

My question is why/how is $\displaystyle E = H \oplus L $???

Can anyone help?

(see my intuitive diagrams - my notion of $\displaystyle H \oplus L $ is a line going through a plane)

Maybe I need to read some basic differential geometry?

Peter