Editing Linear form (section)

== 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 function|surjective]] (that is, its range is all of {{mvar|k}}).
* Indexing into a vector: The second element of a three-vector is given by the one-form <math>[0, 1, 0].</math> That is, the second element of <math>[x, y, z]</math> is <math display=block>[0, 1, 0] \cdot [x, y, z] = y.</math>
* [[Mean]]: The mean element of an <math>n</math>-vector is given by the one-form <math>\left[1/n, 1/n, \ldots, 1/n\right].</math> That is, <math display=block>\operatorname{mean}(v) = \left[1/n, 1/n, \ldots, 1/n\right] \cdot v.</math>
* [[Sampling (signal processing)|Sampling]]: Sampling with a [[Kernel (image processing)|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]], <math>R(t),</math> is given by the one-form <math>w(t) = (1 + i)^{-t}</math> where <math>i</math> is the [[Discount window|discount rate]]. That is, <math display=block>\mathrm{NPV}(R(t)) = \langle w, R\rangle = \int_{t=0}^\infty \frac{R(t)}{(1+i)^{t}}\,dt.</math>

=== Linear functionals in R<sup>''n''</sup> ===
Suppose that vectors in the real coordinate space <math>\R^n</math> are represented as column vectors
<math display=block>\mathbf{x} = \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>

For each row vector <math>\mathbf{a} = \begin{bmatrix}a_1 & \cdots & a_n\end{bmatrix}</math> there is a linear functional <math>f_{\mathbf{a}}</math> defined by
<math display=block>f_{\mathbf{a}}(\mathbf{x}) = a_1 x_1 + \cdots + a_n x_n,</math>
and each linear functional can be expressed in this form.

This can be interpreted as either the matrix product or the dot product of the row vector <math>\mathbf{a}</math> and the column vector <math>\mathbf{x}</math>:
<math display=block>f_{\mathbf{a}}(\mathbf{x}) = \mathbf{a} \cdot \mathbf{x} = \begin{bmatrix}a_1 & \cdots & a_n\end{bmatrix} \begin{bmatrix}x_1\\ \vdots\\ x_n\end{bmatrix}.</math>

=== Trace of a square matrix ===
The [[Trace (linear algebra)|trace]] <math>\operatorname{tr} (A)</math> of a square matrix <math>A</math> is the sum of all elements on its [[main diagonal]]. Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a [[vector space]] from the set of all <math>n \times n</math> matrices. The trace is a linear functional on this space because <math>\operatorname{tr} (s A) = s \operatorname{tr} (A)</math> and <math>\operatorname{tr} (A + B) = \operatorname{tr} (A) + \operatorname{tr} (B)</math> for all scalars <math>s</math> and all <math>n \times n</math> matrices <math>A \text{ and } B.</math>

=== (Definite) Integration ===
Linear functionals first appeared in [[functional analysis]], the study of [[Function space|vector spaces of functions]]. A typical example of a linear functional is [[Integral|integration]]: the linear transformation defined by the [[Riemann integral]]
<math display=block>I(f) = \int_a^b f(x)\, dx</math>
is a linear functional from the vector space <math>C[a, b]</math> of continuous functions on the interval <math>[a, b]</math> to the real numbers. The linearity of <math>I</math> follows from the standard facts about the integral:
<math display=block>\begin{align}
     I(f + g) &= \int_a^b[f(x) + g(x)]\, dx = \int_a^b f(x)\, dx + \int_a^b g(x)\, dx = I(f) + I(g) \\
  I(\alpha f) &= \int_a^b \alpha f(x)\, dx = \alpha\int_a^b f(x)\, dx = \alpha I(f).
\end{align}</math>

=== Evaluation ===
Let <math>P_n</math> denote the vector space of real-valued polynomial functions of degree <math>\leq n</math> defined on an interval <math>[a, b].</math> If <math>c \in [a, b],</math> then let <math>\operatorname{ev}_c : P_n \to \R</math> be the '''evaluation functional'''
<math display=block>\operatorname{ev}_c f = f(c).</math>
The mapping <math>f \mapsto f(c)</math> is linear since
<math display=block>\begin{align}
     (f + g)(c) &= f(c) + g(c) \\
  (\alpha f)(c) &= \alpha f(c).
\end{align}</math>

If <math>x_0, \ldots, x_n</math> are <math>n + 1</math> distinct points in <math>[a, b],</math> then the evaluation functionals <math>\operatorname{ev}_{x_i},</math> <math>i = 0, \ldots, n</math> form a [[Basis of a vector space|basis]] of the dual space of <math>P_n</math> ({{harvtxt|Lax|1996}} proves this last fact using [[Lagrange interpolation]]).

=== Non-example ===
A function <math>f</math> having the [[equation of a line]] <math>f(x) = a + r x</math> with <math>a \neq 0</math> (for example, <math>f(x) = 1 + 2 x</math>) is {{em|not}} a linear functional on <math>\R</math>, since it is not [[Linear function|linear]].<ref group="nb">For instance, <math>f(1 + 1) = a + 2 r \neq 2 a + 2 r = f(1) + f(1).</math></ref> It is, however, [[Affine-linear function|affine-linear]].