Consider a basis ( b 1 , b 2 , … … , b n ) of a vector space V of dimension n over a field F. The definition of a basis implies that every v ∈ ∈ V may be written v = a 1 b 1 + ⋯ ⋯ + a n b n , with a 1 , … … , a n in F, and that this decomposition is unique. The scalars a 1 , … … , a n are called the coordinates of v on the basis. They are also said to be the coefficients of the decomposition of v on the basis. One also says that the n - tuple of the coordinates is the coordinate vector of v on the basis, since the set F n of the n -tuples of elements of F is a vector space for componentwise addition and scalar multiplication, whose dimension is n.