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.

Let F be a field, for example R or C, and let V be a set of objects. These objects are called vectors and are shown in bold such as v. We are interested in defining a special kind of multiplication, called scalar multiplication, and addition.

If s is in F then we insist sv is in V for all v in V. This means 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. In fact, we demand V has an element O and additive inverses so that V is a commutative additive group.

V is almost a vector space over F but we have to insist on a few more conditions related to scalar multiplication...