Tangent spaces to the orbit of a Lie group

The restriction $f_A$ of the map $A$ to $M$ induces a diffeomorphism of $M$, remark $df_A=A$ since $A$ is linear. We have $f_A(x_0)=A(x_0)$, and $df_A(T_{x_0}(M))=A(T_{x_0}M)=T_{f_A(x_0)}M=T_{A(x_0)}M$.