Let $A$ be a linear map of $\mathbb{R}^n$ and $b \in \mathbb{R}^n$. Show that the mapping $x \rightarrow Ax + b$ is a diffeomorphism of $\mathbb{R}^n$ if and only if $A$ is nonsingular.
I am not sure where to start with this proof. I feel like if I assume that $x \rightarrow Ax + b$ is a diffeomorphism then doesn't $A$ being invertible have to hold?
Conversely if I assume $A$ is invertible, I'm not sure how i would show $x \rightarrow Ax + b$ is a diffeomorphism.
Any help in the right direction would be appreciated.
All affine maps are smooth and inverse of $Ax+b$ is $A^{-1}(x-b)$ when it exists. So you only have to show that $x \to Ax+b$ is bijective if and only if $A$ is non-singular. Now $Ax+b$ is bijective if and only if $Ax$ is bijective which is true if and only if $A$ is non-singular.
If $Ax+b$ is a bijection then $Ax+b=y$ has a unique solution $x$ for every $y$. Hence $Ax=z$ has a unique solution $x$ for every $y$. This means $x \to Ax$ is a bijection. A linear map is a bijection if and only if it is non-singular.