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In defining the tangent space of a smooth manifold at a point $p$, one can consider a local coordinate $\phi: U \to \phi(U)$ with $p \in U$ and define the tangent space as the tangent space to $\phi(p)$ in $\phi(U)$ which is an open subset of $\{\mathbb R}^n$. However this definition would depend on the choice of $\phi$, which is undesirable. Now, the idea is to use the natural isomorphisms between tangent spaces thus defined (the derivatives of the change of coordinate maps) to identify these spaces.