Why Is a Fiber A Vector Space

Question

I'm trying to learn about vector bundles in the context of differentiable manifolds and have a very basic question which most introductions to the topic seem to consider obvious. Let $M$ be a manifold and let $E=M\times \mathbb{R}^n$ be the total space. Let $p:E\to M$ be the projection. My question is, for $x\in M$ why is $p^{-1}(x)\subset E$ a vector space?

Answer 1

The set $p^{-1}(x)$ is just $\{x\}\times\mathbb{R}^n$. This has a canonical vector space structure by using the standard vector space structure on the second coordinate. That is, define addition and scalar multiplication by $(x,v)+(x,w)=(x,v+w)$ and $\lambda\cdot (x,v)=(x,\lambda v)$.