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, ${\displaystyle V^{*}}$; other notations are also used, such as ${\displaystyle V'}$, ${\displaystyle V^{\#}}$ or ${\displaystyle V^{\vee }.}$ 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).

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).

Suppose that vectors in the real coordinate space ${\displaystyle \mathbb {R} ^{n}}$ are represented as column vectors $${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}$$

For each row vector ${\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}}$ there is a linear functional ${\displaystyle f_{\mathbf {a} }}$ defined by $${\displaystyle f_{\mathbf {a} }(\mathbf {x} )=a_{1}x_{1}+\cdots +a_{n}x_{n},}$$ 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 ${\displaystyle \mathbf {a} }$ and the column vector ${\displaystyle \mathbf {x} }$: $${\displaystyle f_{\mathbf {a} }(\mathbf {x} )=\mathbf {a} \cdot \mathbf {x} ={\begin{bmatrix}a_{1}&\cdots &a_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}.}$$

The trace ${\displaystyle \operatorname {tr} (A)}$ of a square matrix ${\displaystyle A}$ 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 ${\displaystyle n\times n}$ matrices. The trace is a linear functional on this space because ${\displaystyle \operatorname {tr} (sA)=s\operatorname {tr} (A)}$ and ${\displaystyle \operatorname {tr} (A+B)=\operatorname {tr} (A)+\operatorname {tr} (B)}$ for all scalars ${\displaystyle s}$ and all ${\displaystyle n\times n}$ matrices ${\displaystyle A{\text{ and }}B.}$

Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral $${\displaystyle I(f)=\int _{a}^{b}f(x)\,dx}$$ is a linear functional from the vector space ${\displaystyle C[a,b]}$ of continuous functions on the interval ${\displaystyle [a,b]}$ to the real numbers. The linearity of ${\displaystyle I}$ follows from the standard facts about the integral: $${\displaystyle {\begin{aligned}I(f+g)&=\int _{a}^{b}[f(x)+g(x)]\,dx=\int _{a}^{b}f(x)\,dx+\int _{a}^{b}g(x)\,dx=I(f)+I(g)\\I(\alpha f)&=\int _{a}^{b}\alpha f(x)\,dx=\alpha \int _{a}^{b}f(x)\,dx=\alpha I(f).\end{aligned}}}$$