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Reflections and Reflection Groups - Basic Geometry

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

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Re: Reflections and Reflection Groups - Basic Geometry

this has nothing to do with differential geometry (we're not considering differentiability of anything). where you're going wrong is in thinking about which direction L is going. L is perpendicular to the hyperplane H, in physics, one often hears the term "normal vector (at 0)" to describe L. in other words, L is coming "straight up" (perpendicular) out of the plane. for example, if H was the xy-plane in euclidean 3-space, then L would be the z-axis.

Plane + perpendicular line = 3-space.

actually, we could use ANY line, not lying in the plane H but then we lose orthogonality. and orthogonality makes the arithmetic easier (it makes our basis for V look like the standard basis "rotated" to line up with H and L).

in other words, one way to describe a point in 3-space, is to pick a point in a plane (any plane will do, imagine it being the xy-plane "tilted" by some amount), and to pick a distance "up off the plane" (how far along the line L we've come).

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Re: Reflections and Reflection Groups - Basic Geometry

Yes, take your point re differential geometry - my comment was rather careless ... was thinking that a differential geometry text may cover some of this in the preamble to winding into the subject ... but I suppose an algebra text covering vector spaces would be more likely to help!

Thanks for help around this roadblock ... not quite sure why I did not see it ...

Can now take Reflections and Refection Groups a bit further thanks to your help

Peter