Linear Functional

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the matrix product with the row vector on the left and the column vector on the right. In general, if V is a vector space over a field k, then a linear functional ƒ is a function from V to k, which is linear:

for all

for all

The set of all linear functionals from V to k, Hom_k(V,k), is itself a vector space over k. This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space. It is often written V* or when the field k is understood.