Soft question:Intuition for tangent space

Question

I am trying to learn topology, from An introduction to manifolds of Loring W.Tu second edition to be precise, while I saw the definition of tangent space I don't think I understood what they are and what they mean, how would you explain the concept of tangent space in an informal manner.

Answer 1

Below I give the notion as in Differential Geometry, other definitions of course exist:

For simplicity let us consider our manifold to be the surface of the unit sphere in $\mathbb{R}^3$, which of course may be described as: S = $\{ a \in \mathbb{R^3} : ||a|| = 1\}$

$v \in \mathbb{R}^3$ is a tangent vector of $S$ at $x \in S$ if there is a smooth curve, $\gamma:\mathbb{R} \rightarrow S$ s.t.

1. $\gamma(0) = x$
2. $\dot{\gamma}(0) = v $

The set of tangent vectors of $S$ at $x$, denoted $T_x(S)$ is given to be: $$T_x(S) = \{\dot{\gamma}(0)| \gamma \textrm{ smooth,}\gamma(0) = x\}. $$

Say $x = (1,0,0)$ Then, intuitively, what would the tangent space be?
what will its dimension be?

The intuition we develop in 3 space will hold for $n$ space