I start reading about matrix manifolds and in an introduction section it is mentioned the statement below. Is it possible to provide an intuition in terms of basic (e.g. fundamental sub-spaces) linear algebra terms?
An $n \times n$ matrix $A$ naturally induces a mapping on $Grass(p,n)$ defined by $$\mathcal{S}\in Grass(p,n) \mapsto A\mathcal{S}:=\{Ay : y \in \mathcal{S} \}$$ where $Grass(p,n)$ admits a structure of manifold called the Grassmann manifold.
Thank you in advance!
The statement is actually incorrect. Unless $\det A\ne0$, the image of $S$ under $A$ may have smaller dimension, and therefore "fall out" of the Grassmannian manifold being considered. If $A$ is invertible, then it induces a linear isomorphism of the ambient space and the image of $S$ will have the same dimension as $S$.