5.2 Vector spaces

The last section introduced several ideas about vector spaces using familiar notions from R² and C. It’s time to generalize. vector space vector

Let F be a field, for example, Q, R, or C. Let V be a set of objects which we call vectors. We display vectors in bold such as v. F`bold

We are interested in defining vector addition and a special kind of multiplication called scalar multiplication. If s is in F, then we insist s v is in V for all v in V. The set V is closed under multiplication by scalars from the field F. While V may have some kind of multiplication defined between its elements, we do not need to consider it here.

For any v₁ and v₂ in V, we also insist v₁ + v₂ is in V and that the addition is commutative. Thus, V is closed under addition. V must have an identity element 0 and additive inverses so that V is a commutative additive group. group

V is almost a vector space over F, but we insist on a few more...