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Submanifolds, Matrix groups

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the set of all 2x2 matrices of rank 1 is a submanifold of R^4

    2. Relevant equations

    3. The attempt at a solution

    The hint in the book was to show that the determinant function is a submersion on the manifold of nonzero 2x2 matrix M(2) - 0. This is easy to show. So I have that det^{-1}(0) \subset M(2) - 0 is a 3 dimensional sub manifold of M(2) - 0. But how do I show that it's a submanifold of R^4?

    I know that M(2) - 0 is an open subset of R^4... I get the intuitive idea, but I don't see how to write a rigorous proof. How do I show that the set of 2x2 matrices of rank 1 is a submanifold of R^4 if I just showed that it is a submanifold of M(2) - 0?
  2. jcsd
  3. Feb 9, 2010 #2
    Think geometrically -- what configuration would be "bad", that is, cause your manifold not to be a submanifold of [tex]\mathbb{R}^4[/tex]?
  4. Feb 9, 2010 #3
    That didn't help me too much, I just looked up the definition of submanifold (explicitly) and just used that. It works quite nicely.
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