 Solution

A tangent vector to $M$ at $p$ is a velocity ${\gamma }^{\prime }(0)$ of a smooth curve $\gamma :(-\epsilon ,\epsilon )\to M$ with $\gamma (0)=p$. The tangent space is the set of all such velocities:It is a $k$-dimensional linear subspace of ${\mathbb{R}}^n$; for a local parametrization $\varphi$, it is the image of the injective derivative $D\varphi$ at the corresponding parameter point.