Definition talk:Vector Space Axioms

Scalar multiplication
I think I am missing something. Let $\mathbf x \in G$ and $\lambda \in K$. Which axiom says that $\lambda \circ \mathbf x \in G$? This sounds trivial, but I want to make sure that this fact is not hidden behind some terminology.--Julius (talk) 21:07, 11 February 2021 (UTC)


 * Heh! Took me a while to find it, this approach was written long ago and I have basically forgotten it.


 * This is the definition of vector space $\struct {G, +_G, \circ}_K$:


 * $G$ is a set of objects, called vectors.


 * $+_G: G \times G \to G$ is a binary operation on $G$


 * $\struct {K, +, \cdot}$ is a division ring whose unity is $1_K$


 * $\circ: K \times G \to G$ is a binary operation.


 * It's that last line: $K \times G \to G$: we have that $\lambda \in K$ and $\mathbf x \in G$, so it follows that $\lambda \circ \mathbf x \in G$. --prime mover (talk) 23:13, 11 February 2021 (UTC)


 * So it was there all along. Thanks.--Julius (talk) 07:58, 12 February 2021 (UTC)