(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Submanifolds, Matrix groups

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