i am looking for a confirmation of the following Lemma as well as a reference:
Let $M \subset \mathbb{R}^m $ be a smooth-manifold and $A \in \mathbb{R}^{n\times m}$ be a full rank matrix with $n\geq m$. Define $$ N = \{ Ax \, |\, x \in M\}. $$ Then N is also a smooth manifold and $$ A\mathcal{T}_x M := \{Av \, | \, v \in \mathcal{T}_x M\} = T_{Ax} N. $$
Is this correct? I would be very glad if someone could point me to a reference since unfortunatly i could not find anything that seems to be directly related. Thanks in advance.
I don't know a reference offhand, but the statements are correct and their proofs is routine.
Hints.
The fact that $N$ is a smooth manifold follows from, e.g., the Inverse Function Theorem, or just as well the local normal form for smooth, constant-rank maps.
To check the claim about tangent spaces, fix any base point $x \in M$ and tangent vector $Y \in T_x M$. Then, there is a curve $\gamma : I \to M$ for some interval $I$ containing $0$ such that $$\gamma(0) = x \qquad \textrm{and} \qquad \gamma'(0) = Y .$$ Now, unwind the definition of the pushforward $T_x A \cdot Y$ in terms of the curve $\gamma$ to conclude that $T_x A \cdot Y \in T_{A x} N$, and use the injectivity of $A$ to show that $T_x A$ is in fact an isomorphism $T_x M \to T_{A x} N$ (for all $X$).