(Or if you prefer: Why are things defined this way?) I noticed that, in my book's definition, scalar multiplication (SM) on vector spaces lacks two familiar things: commutativity and inverses.(adsbygoogle = window.adsbygoogle || []).push({});

The multiplicative inverse concept doesn't seem to apply to SM. Can it? I can't imagine how it could because, for one thing, the multiplicative identity is a scalar and the product of SM is a vector. (Right? I can't find a definition that actually says that 1 is the identity, but that's what I take1to mean. And is 1 meant to just be the multiplicative identity of the set over which the vector space is defined, whatever it happens to be?)v=vfor allvin V

I guess that SM isn't required to be commutative because you want to be able to define vector spaces over different kinds of sets? But isn't SM commutative on vector spaces over fields? That is, for example, if F^{n}is a field, a is in F, and (x_{1}, ..., x_{n}) is in F^{n}, then I could define SM' as

a(x_{1}, ..., x_{n}) = (ax_{1}, ..., ax_{n}) = (x_{1}, ..., x_{n})a

and as long as everything else holds, F^{n}with SM' would be a vector space? Is commutativity for SM interpreted in another way? For example

ab(v) = ba(v)

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# Why is scalar multiplication on vector spaces not commutative?

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