- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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 [itex] H \subset E [/itex] through the origin, let L = the line through the origin that is orthogonal to H. So [itex] E = H \oplus L [/itex]"

My question is why/how is [itex] E = H \oplus L [/itex]?

Can anyone help?

(see my intuitive diagrams - my notion of [itex] H \oplus L [/itex] is a line going through a plane)

Maybe I need to read some basic differential geometry?

Peter

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 [itex] H \subset E [/itex] through the origin, let L = the line through the origin that is orthogonal to H. So [itex] E = H \oplus L [/itex]"

My question is why/how is [itex] E = H \oplus L [/itex]?

Can anyone help?

(see my intuitive diagrams - my notion of [itex] H \oplus L [/itex] is a line going through a plane)

Maybe I need to read some basic differential geometry?

Peter

#### Attachments

Last edited: