1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Topology- Hyperplane proof don't understand

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex] \mathbf {a} \in R^n [/tex] be a non zero vector, and define {[tex] S = \mathbf {x} \in R^n : \mathbf {a} \cdot \mathbf {x} = 0 [/tex] }. Prove that S interior [tex] = {\o} [/tex]

    2. Relevant equations

    3. The attempt at a solution

    Intuitively I understand that if a is a vector in R^3, S would be a plane. And if I place an open ball on any point on the plane it would include points that are not on the plane. So S interior would be empty.

    But I'm not sure how to prove this rigorously?
    I'm supposed to somehow use the hint that a hyperplane in R^n will have 1 dimension less than R^n (like the plane in R^3 has 2 dimensions)
    But I am confused about the dimension of the plane? All vectors on a plane in R^3 will have 1 (the same) component = 0? So does this mean that the dimension of the plane is 2?
  2. jcsd
  3. Sep 29, 2010 #2
    Can I just show that there is a point in the ball that has n components ?
    Instead of n-1
  4. Sep 29, 2010 #3


    User Avatar
    Science Advisor
    Homework Helper

    If a.x=0 you can pretty easily find a point y in any ball around x such that a.y is not zero.
  5. Sep 30, 2010 #4
    I'm a little confused about that.
    Do I need to differentiate between a point and a vector?
    Like if I show that there are points in the ball that are not on the plane, then do I need to specify that the vector corresponding to that point is not perp a ?
  6. Sep 30, 2010 #5


    User Avatar
    Science Advisor
    Homework Helper

    Now I'm confused about that. Any point y in the ball around x of radius r is given by y=x+a where 'a' is any vector whose length is less than r. An awful lot of them don't satisfy a.y=0. Can you find just one?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook