Why is the set of tangent vectors at 0 in R^m bijective with R^m itself?

Question

Barden & Thomas write in An Introduction to Differentiable Manifolds that the set of tangent vectors at 0 in R^m is bijective with R^m?

Why is this the case? Can 0 be replaced by an arbitrary point p of R^m?

Answer 1

Intuitively:

The tangent vector space of a line ($\mathbb{R^1}$) at a point of the line (that can be the origin) is the line itself.

The tangent space of a plane($\mathbb{R^2}$) is the plane itself.

ans so for $\mathbb{R^n}$.....

Any formal definition of the tangent vector space of a variety conserve this intuition.

Answer 2

Yes you can replace $0$ with any $p\in\mathbb R^m$.

By definition the tangent space, $T_pM$, is an $m$-dimensional vector space. But any two $m$-dimensional vector spaces are isomorphic.