Tangent space from a local parametrization

If $\varphi :U\subseteq {\mathbb{R}}^d\to M\subseteq {\mathbb{R}}^n$ is a local parametrization with $\varphi (u)=p$, thenThe derivative has rank $d$, so this is a $d$-dimensional vector subspace. A change of parametrization composes $D{\varphi }_u$ with an invertible coordinate-change derivative, leaving its image unchanged.