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*