What is the Submanifold of Rank 1 2x2 Matrices in R^4?

[Geometrick]

Homework Statement

Show that the set of all 2x2 matrices of rank 1 is a submanifold of R^4

Homework Equations

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?

 

[ystael]

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

 

[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.