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The tangent space at a point $p$ is defined as the set of ordered pairs $(p,v)$ where the second entry $v$ ranges over all of $\mathbb{R}^n$. The first slot records the basepoint and the second slot records the vector part.

Conditions: $p \in \mathbb{R}^n$; $v \in \mathbb{R}^n$

The tangent space at a point $p$ is defined as the set of ordered pairs $(p,v)$ where the second entry $v$ ranges over all of $\mathbb{R}^n$. The first slot records the basepoint and the second slot records the vector part.

Conditions: $p \in \mathbb{R}^n$; $v \in \mathbb{R}^n$