Partial derivatives of all orders of linear map exist

Question

If F is a linear map from R^n to R^m is it true that F is C^infinity, i.e. partial derivatives of all orders exist? My thought is that the answer should be "yes," because the derivative of F is just F itself... But I have been unable to deduce anything from this. This is not homework, and stems from my desire to solve an exercise in Guillemin and Pollack (i.e. to show that any k-dimensional vector subspace of R^N is diffeomorphic to R^k). It seems like I am missing something very basic.

Answer 1

Yes this is true as the derivative of a linear map $F$ at a point $a \in \mathbb R^n$ is a linear map equal to $F$.

Therefore the derivative $F^\prime$ is the constant function that maps $x \in \mathbb R^n$ to the constant linear map $F$. Hence the second derivative is equal to $0$ as all further order derivatives.