Q1: Let $M$ be a smooth manifold of dimension $n$. What is the dimension of $T_p(M)$?

A1: $n$

A2: 0

A3: $n^2$

A4: $n-1$

Q2: Let $M$ be a smooth manifold of dimension $n$. What is a basis of $T_p(M)$?

A1: $(\varphi_{\ast} (e_1), \ldots, \varphi_{\ast} (e_n) )$

A2: $(e_1, \ldots, e_n)$

A3: $(h (e_1), \ldots, h(e_n))$

Q3: Let $M$ be a smooth submanifold in $\mathbb{R}^n$ of dimension $k$ . What is a basis of $T^{\textrm{sub}}_p(M)$?

A1: $(\frac{\partial \varphi}{\partial x_1}(\tilde{p}) \ldots, \frac{\partial \varphi}{\partial x_n}(\tilde{p}))$

A2: $(e_1, \ldots, e_n)$

A3: $(h (e_1), \ldots, h(e_n))$