Linear Forms

Let $n$ be a positive a positive integer. A function

$$f:{\mathbb{R}}^n\to \mathbb{R}$$

is a linear form (or linear functional) if there is a row-vector

$$\mathbf{a}=\left[\begin{array}{cccc}{a}_1& a_2& \dots & a_n\end{array}\right],a_i\in \mathbb{R},i=1,2,\dots n$$

such that for any column vector $\mathbf{x}={\left[\begin{array}{cccc}{x}_1& x_2& \dots & x_n\end{array}\right]}^T\in {\mathbb{R}}^n$, we have

$$\begin{array}{rl}f(\mathbf{x})& =\mathbf{ax}\\ & =\left[\begin{array}{cccc}{a}_1& a_2& \dots & a_n\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]\\ & =a_1{x}_1+a_2{x}_2+a_n{x}_n\end{array}$$