I just had a short look on the definition of a Vector Space and couldn't find any obvious reason why
$\lambda a = a \lambda$
where $\lambda$ is an element of the field $K$ and $a$ is an Element of the set $M$, should be true. Its kind of intuitive since all the vectorspaces I normally come across have this property (Polynoms, indefinitly diffrentiable functions, the $\mathbb{R}^n$). However since $\lambda \in K$ and and $a \in M$, I see no justification for this product being commutative. Is this dependend on the definition of the product between the field and the set? Or is this generally true? If not it would be great if you could provide some examples.
Thanks in advance.