A vector space is an object between a field and a group; it has two ops—addition and scalar multiplication. Its not quite a field and its more than a group. constituents A set V An addition on V An scalar multiplication on V such that… requirements commutativity in add.: u+v=v+u associativity in add. and mult.: (u+v)+w=u+(v+w); (ab)v=a(bv): \forall u,v,w \in V and a,b \in \mathbb{F} distributivity: goes both ways a(u+v) = au+av AND!! (a+b)v=av+bv: \forall a,b \in \mathbb{F} and u,v \in V additive identity: \exists 0 \in V: v+0=v \forall v \in V additive inverse: \forall v \in V, \exists w \in V: v+w=0 multiplicative identity: 1v=v \forall v \in V additional information Elements of a vector space are called vectors or points. vector space “over” fields Scalar multiplication is not in the set V; instead, “scalars” \lambda come from this magic faraway land called \mathbb{F}. The choice of \mathbb{F} for each vector space makes it different; so, when precision is needed, we can say that a vector space is “over” some \mathbb{F} which contributes its scalars. Therefore: A vector space over \mathbb{R} is called a real vector space A vector space over \mathbb{C} is called a real vector space