Dynamical Systems on a Manifold

Question

My textbook on dynamical systems says that a dynamical system written as:

$\mathbf{x'} = \mathbf{f(x)}$,

where the right-hand-side $\mathbf{f(x)}$ is a vector field defined on a manifold equivalent to the dimension of the system. My question is that if this is a linear system, where $\mathbf{f(x)} = A\mathbf{x}$, is this still defined on a manifold?

Thanks.

Thomas

Answer 1

Sure; in particular note that just a finite dimensional vector space is a manifold as well. Its tangent spaces are isomorphic to the original space (intuitively, the isomorphism amounts to "moving the origin" to the point where you take the tangent space).