How is the tangent space $T_p\mathbb{R}^n$ defined in this clip?

The basepoint $p$ remains important because it determines the linear map $Df(p)$ used in the output pair. The derivative depends on the point at which it is evaluated.

Conditions: Applies to the basepoint derivative $df_p(v) = (q, Df(p)v)$.

A vector field over $\mathbb{R}^3$ assigns to each point $p$ a vector $v$ belonging to the tangent space $T_p\mathbb{R}^3$ at that point.

Conditions: The ambient space in this example is $\mathbb{R}^3$.

The basepoint derivative $df_p$ is a map from the tangent space at $p$ to the tangent space at $q=f(p)$. It sends a tangent vector $v$ to the pair $(q, Df(p)v)$, packaging the ordinary derivative with the target basepoint.

Conditions: $U$ is an open subset of $\mathbb{R}^n$.; $f : U \to \mathbb{R}^m$ and $f \in C^1$.; $q = f(p)$.

The point $p$ serves as a basepoint. Retaining it motivates a basepoint-compatible notion of derivative, which is discussed next in the video.

Conditions: Applies to the first coordinate $p$ in $(p,v)\in T_p\mathbb{R}^n$

The cotangent space at $p$ is the dual space of the tangent space at $p$. It is written as $\operatorname{Hom}(T_p\mathbb{R}^n, \mathbb{R})$ because it consists of all linear maps from the tangent space $T_p\mathbb{R}^n$ to the field $\mathbb{R}$.

Conditions: $p$ is a basepoint.