TITLE: How to prove $GL^+ (2, \mathbb{R})$ is connected
QUESTION [5 upvotes]: I am looking to prove that the set of $2\times 2$ matrices with positive determinant is connected. I understand that the set of invertible $2\times 2$ matrices is however disconnected since the determinant can not equal zero. 
I know That my aim is to prove that $GL^+ (2,\mathbb{R})$ cannot be written as a disjoint Union of two open and non-empty subsets. How do I go about this? 
Thanks

REPLY [4 votes]: The Gram-Schmidt process allows us to make a deformation retract from $GL^+(2,\mathbb{R})$ to $SO(2)$. Since $SO(2) \simeq S^1$, it follows that $SO(2)$ is path-connected and thus $GL^+(2,\mathbb{R})$ must also be.
This argument can be generalized to show that $GL^+(n,\mathbb{R})$ is connected, using the fact that $SO(n)$ is connected (which can be inferred, for instance, from the fact that the spheres are connected, but this is  a separate argument).