B. Vector Spaces.
If V is an additive abelian group with elements A, B, . . . , F a field with elements a, b, . . . , and if for each a f F and A * V the product aA denotes an element of V, then V is called a (left) vector space over F if the following assumptions hold:
The reader may readily verify that if V is a vector space over then oA = 0 and aO = 0 where o is the zero element of F and 0 that of V. For example, the first relation follows from the equations:
Sometimes products between elements of F and V are written in the form A a in which case V is called a right vector space over F to distinguish it from the previous case where multiplication by field elements is from the left. If, in the discussion, left and right vector spaces do not occur simultaneously, we shall simply use the term "vector space."