This is a really basic question but I always wanted to know the answer to it. If we have a vector space $V$ over a field $K$, then scalar multiplication is usually defined by taking $\alpha v$, $\alpha \in K$ and $v \in V$ and the scalar always come on the left side. I know that in this case one can freely change the order of multiplication $\alpha v = v\alpha$. However, what axiom of vector space justifies this exchange? Since scalar multiplication is always introduced by placing the scalar on the left, how can we even start to consider a scalar on the right? Shouldn't we have some property about, say, $1v = v1 = v$? I have never found a reference in which some property is explicitly stated.
Right and left multiplication by scalar on a vector space
linear algebravector-spaces
Best Answer
Well, in some sense the order doesn’t matter, because these are the objects of different nature! When we are working with operation $*:A\times A\to B$, the order is important because both $\forall a_{1},a_{2} \in A: a_{1}*a_{2}\in B \text{ and } a_{2}*a_{1}\in B$ by definition. In your case if $V$ is a vector space over $\mathbb{K} $ field, then multiplication by number is defined like \begin{gather} \mathbb{K}\times V \to V\\ (\alpha ,v)\mapsto \alpha v.\end{gather} The $v\alpha$ notation is in some sense meaningless, because $\mathbb{K}$ and $V$ are principally different sets. You can always define operation \begin{gather} \mathbb{K}\times V \to V\\ (\alpha ,v)\mapsto v\alpha.\end{gather} But it is some different operation. If it does have the same properties as $\alpha v$, then there is an isomorphism of structures, which allows you to rearrange vectors and scalars.