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."