Reflections and Reflection Groups - Basic Geometry

[Math Amateur]

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

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

Can anyone help?

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

Maybe I need to read some basic differential geometry?

Peter

 

[DonAntonio]

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

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


*** Without the basic definitions I must rely on the standard ones, and then H is a hyperplane = a maximal proper subspace of E = the kernel of some non-zero lin. functional = a subspace of dimension n - 1 if dim E = n.

Thus, H is a hyperplane iff $E=H\oplus Span\{v\}$ , for any $v\notin H$ , and this is basic (not necessarily finite-dimensional) linear algebra, no differential geometry needed at all.

DonAntonio



Can anyone help?

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

Maybe I need to read some basic differential geometry?

Peter

...

 

[HallsofIvy]

If H is a "hyperplane" in space E through the origin, E having dimension n, then H is a subspace of E of dimension n-1. There exist a basis for H consisting of n- 1vectors which can be extended to a basis for E by adding one more vector perpendicular to all the n-1 basis vectors in H. The space spanned by that one vector is L. every vector in E can then be written as a linear combination of the basis vectors for H, and so is in H, and a vector in L. That is essentially what "E= H⊕L" means.

 

[Math Amateur]

Thanks for the help!

Thanks to your help, now over that "roadblock"

Peter

 

[blue_raver22]

Hello Peter,

Thank you for sharing your question and diagrams. I can understand your confusion with the notation and definition of reflections in Kane's book. To clarify, the notation E = H \oplus L means that the vector space E is the direct sum of the two subspaces H and L. In other words, every vector in E can be uniquely expressed as a sum of a vector in H and a vector in L. This is a fundamental concept in linear algebra.

In the context of reflections, this means that any vector in the vector space E can be written as a sum of a vector in the hyperplane H and a vector in the line L. This is because reflections are defined as transformations that flip a vector across a hyperplane or a line, while keeping the hyperplane or line fixed. Therefore, the vector space E is divided into two parts - the hyperplane H and the line L - where each part represents the fixed and moving components of the reflection transformation.

I hope this explanation helps clarify the notation and concept of reflections and reflection groups in basic geometry. If you are still having difficulties, I would recommend reviewing some basic linear algebra concepts, such as vector spaces and direct sums, before continuing with Kane's book. Good luck with your studies!