Vector Space over Field F:
An association of scalars a, b, c.... from a field F with Vectors A, B, C... (undefined) from an Abelian group G.
The association has certain properties, for ex, aA is defined and belongs to G.
EDIT: A typical question might be, is multiplication (rule of association) of a Matrix (Vector) by a scalar from a field F (real or complex) a Vector Space?
1) Give the scalar and define the matrix.
2) Are the matrices an Abelian Group?
3) Are the rules of association satisfied? Especially does aA belong to G?