Let $F:\mathbb{R}^m \to \mathbb{R}^n$ where $n \neq m$. I'm confused as to whether this map can be invertible or not.
The Inverse Function Theorem ensures the existence of a local inverse provided that the Jacobian determinant is non-zero. But clearly this is only defined for maps between vector spaces of the same dimension.
So in the case where $n \neq m$ the Inverse Function Theorem says nothing. Does this necessarily mean that the map can't be invertible?