= Solution
An embedded $k$-dimensional <submanifold> $M\subseteq\mathbb R^n$ is locally a coordinate $k$-plane: for every $p\in M$, there are open neighbourhoods $U\ni p$, $V\subseteq\mathbb R^n$ and a <diffeomorphism> $\Phi:U\to V$ with
$$
\Phi(M\cap U)=V\cap(\mathbb R^k\times\{0\}^{n-k}).
$$
For the present $C^1$ setting, $\Phi$ and its inverse are continuously differentiable; in the smooth convention both are smooth.
A <tangent vector> to $M$ at $p$ is a <velocity> $\gamma'(0)$ of a continuously differentiable curve $\gamma:(-\epsilon,\epsilon)\to M$ with $\gamma(0)=p$. The <tangent space> is the collection of all these vectors:
$$
\boxed{T_pM=\{\gamma'(0):\gamma(0)=p,\ \gamma\text{ is a }C^1\text{ curve in }M\}.}
$$
In the local coordinates above, it is $D\Phi(p)^{-1}(\mathbb R^k\times\{0\}^{n-k})$, so it is a $k$-dimensional linear subspace of $\mathbb R^n$.
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