The tangent space of a manifold at a point given as the kernel of the jacobian of a submersion
Solution 1:
You are right with the inclusion $T_p(V)\subset \ker(\phi_*)$. Just take $v\in T_p(V)$ and let $\alpha(t)$ be a curve in $V$ such that $\alpha(0)=p$ and $\alpha'(0)=v$ , then we have that
$$\phi_*(v)=\frac{d}{dt}\Big|_{t=0}\phi(\alpha(t))=\frac{d}{dt}\Big|_{t=0}q=0.$$
Finally, we get the equality $T_p(V)=\ker(\phi_*)$ by dimension counting.
Solution 2:
Hint: Let $j : V \to M$ be the canonical injection. Show that $dj(p) : T_pV \to T_pM$ induces an isomorphism between $T_pV$ and $\mathrm{ker} (d\phi(p))$.