Linear form

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 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).

Examples The constant zero function, mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is surjective (that is, its range is all of k).

Indexing into a vector: The second element of a three-vector is given by the one-form [ 0 , 1 , 0 ] . {\displaystyle [0,1,0].} That is, the second element of [ x , y , z ] {\displaystyle [x,y,z]} is [ 0 , 1 , 0 ] ⋅ [ x , y , z ] = y . {\displaystyle [0,1,0]\cdot [x,y,z]=y.}

Mean: The mean element of an n {\displaystyle n} -vector is given by the one-form [ 1 / n , 1 / n , … , 1 / n ] . {\displaystyle \left[1/n,1/n,\ldots ,1/n\right].} That is, mean ⁡ ( v ) = [ 1 / n , 1 / n , … , 1 / n ] ⋅ v . {\displaystyle \operatorname {mean} (v)=\left[1/n,1/n,\ldots ,1/n\right]\cdot v.}

Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location. Net present value of a net cash flow, R ( t ) , {\displaystyle R(t),} is given by the one-form w ( t ) = ( 1 + i ) − t {\displaystyle w(t)=(1+i)^{-t}} where i {\displaystyle i} is the discount rate. That is, N P V ( R ( t ) ) = ⟨ w , R ⟩ = ∫ t = 0 ∞ R ( t ) ( 1 + i ) t d t . {\displaystyle \mathrm {NPV} (R(t))=\langle w,R\rangle =\int _{t=0}^{\infty }{\frac {R(t)}{(1+i)^{t}}}\,dt.}

Linear functionals in Rn Suppose that vectors in the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} are represented as column vectors

x = [ x 1 ⋮ x n ] . {\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}

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