Prove the equality of the following tangent spaces

Question

Let $M$ be a n-dimensional manifold and $q\in M$. Let $W_q$ be an open neighborhood of $0$ in $T_qM$ such that $\exp_{q\ |W_q}:W_q\to \exp_q(W_q)$ is a diffeomorfism. Consider $E$ a subvector-space of $T_qM$ and define $M'=\exp_q(W_q\cap E)$. Clearly $M'$ is a submanifold of $M$.

Question:

$T_qM'$ is equal to $E$?

Answer 1

Yes, that's correct, since for $c(t)=\exp_q(tv)$ you have $c^\prime(0) =v$ (it's by definition the geodesic through $q$ with initial vector $v$. Now choose $v\in T_qM^\prime$.