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Reflections and Reflection Groups - Basic Geometry

  1. Mar 30, 2012 #1
    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
     

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    Last edited: Mar 30, 2012
  2. jcsd
  3. Mar 30, 2012 #2
    ....
     
  4. Mar 30, 2012 #3

    HallsofIvy

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    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.
     
  5. Mar 30, 2012 #4
    Thanks for the help!

    Thanks to your help, now over that "roadblock"

    Peter
     
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