Vector space

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7/7/2026, 9:26:14 AM · Ancient Tree

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1A vector space is...

1A vector space over a [[Field|field]] $\mathbb{K}$ is a set $E$ equipped with two operations :

2- Vector addition $+:E\times E \rightarrow E$

3- Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$

5such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:

61) $E$ is an [[Abelian group|abelian group]] with $+$.

72) Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$

83) Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$

94) Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$

105) Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.

7/2/2026, 1:17:44 PM · Ancient Tree

Concept created

A vector space is...