# Submanifolds, Matrix groups

1. Feb 8, 2010

### Geometrick

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. Feb 9, 2010

### ystael

Think geometrically -- what configuration would be "bad", that is, cause your manifold not to be a submanifold of $$\mathbb{R}^4$$?

3. Feb 9, 2010

### Geometrick

That didn't help me too much, I just looked up the definition of submanifold (explicitly) and just used that. It works quite nicely.