Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Leaving vector space stable

  1. Sep 23, 2007 #1
    If [itex]G\subset \textrm{End}(V)[/itex], and [itex]W\subset V[/itex] is a subspace of a vector space V, and somebody says "G leaves W stable", does it mean [itex]GW=W[/itex] or [itex]GW\subset W[/itex] or something else?
  2. jcsd
  3. Sep 24, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    It means it maps W to itself.
  4. Sep 24, 2007 #3

    Chris Hillman

    User Avatar
    Science Advisor

    Finite dimensions

    Exercise: what can you say about these alternatives if V is finite dimensional?

    If you read the "What is Information Theory?" thread: Exercise: suppose we have some infinite group G acting on some countably infinite set X. Suppose that for some [itex]A \subset X[/itex], some [itex]g \in G[/itex] takes [itex]A \mapsto B \subset A[/itex]. What is the corresponding statement about stabilizers? If you have read Stan Wagon, The Banach-Tarski Paradox, what does this remind you of? Can you now give a concrete example (with illustration) of this phenomenon? (Hint: hyperbolic plane.)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook