> [!NOTE] Definition (Real Vector Space) > A real vector space is a [[Vector spaces|vector space]] $(V,+,\circ)$ over the [[Real numbers|the set of real numbers]] $\mathbb{R}$: that is, a real vector space is an [[Algebraic Structure|algebraic structure]] $(V,+_{V},\circ),$ where $V$ is a [[Sets|set]], $+_{V}:V\times V\to V$ is a [[Binary Operation|binary operation]] on $V$ and $\circ:\mathbb{R}\times V\to \mathbb{R}$ is a [[Binary Function|binary function]] that satisfies the following conditions: > > (V0) Closure under vector addition: for all $u,v\in V,$ $u+v\in V.$ > (V1) Identity for vector addition: there exists $0_{V}\in V$ so that for all $v\in V,$ $0_{V}+v=v=v+0_{V}.$ > (V2) Associativity of vector addition: for all $u,v,w\in V,$ $(u+v)+w=u+(v+w).$ > (V3) Inverses under vector addition: for all $v\in V,$ there exists $-v\in V$ so that $v+(-v)=0_{V}.$ > (V4) Commutativity of vector addition: for all $u,v\in V,$ $u+v=v+u.$ > > (V5) Distributivity over vector addition: for all $v,w\in V$ and $\lambda\in \mathbb{R},$ $\lambda \circ(v+_{V}w)=\lambda \circ v+_{V}\lambda \circ w$ > (V6) Distributivity over scalar addition: for all $\lambda,\mu\in \mathbb{R}$ and $v\in V,$ $(\lambda+\mu)\circ v=\lambda v+_{V}\mu w$ > (V7) Associativity of scalar multiplication: for all $\lambda,\mu\in\mathbb{R}$ and $v\in V,$ $\lambda \circ (\mu \circ v)=(\lambda \circ \mu)\circ v.$ > (V8) Identity for scalar multiplication: for all $v\in V,$ $1\circ v=v.$ **Note**: We may summarise (V0)-(V4) by noting that $(V,+)$ is an [[Groups|Abelian group]]. > [!Example] >By [[Real n-Space is Finite Dimensional Real Vector Space]], any [[Real n-Space|real n-space]] $\mathbb{R}^{n}$ forms a real vector space > > By [[Real Matrices Form Real Vector Space]], for all $m,n$ positive integers, the set of all $m\times n$ [[Real Matrices|matrices with real entries]] form a real vector space. # Properties **First consequences of axioms**: By [[Scalar Multiplication by Zero in Real Vector Space]], ... By [[Scalar Multiple of Zero of Real Vector Space]], ... By [[Scalar Multiplication by -1 gives Additive Inverse in Real Vector Space]], ...