Linear map from a vector space to its field of scalars
In mathematics, a linear form (also known as a linear functional,[1] a one-form, or a covector) is a linear map[nb 1] 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),[2] or, when the field k is understood, ;[3] other notations are also used, such as ,[4][5] or [2] When vectors are represented by column vectors (as 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).
^Axler (2015) p. 101, §3.92
^ abTu (2011) p. 19, §3.1
^Katznelson & Katznelson (2008) p. 37, §2.1.3
^Axler (2015) p. 101, §3.94
^Halmos (1974) p. 20, §13
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