Linear functionals

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If V is a vector space over a field k, the set of all linear functionals from V to k is itself a vector space over k with addition and scalar multiplication defined pointwise. This space is called the dual space of V, or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted Hom(V, k), or, when the field k is understood,




V

∗




{\displaystyle V^{*}}
; other notations are also used, such as




V
′



{\displaystyle V'}
,




V

#




{\displaystyle V^{\#}}
or




V

∨


.


{\displaystyle V^{\vee }.}
When vectors are represented by column vectors (as it is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).

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