Linear Forms

Definition

Let $V$ be a $n$dimensional vector space. For a given constant $a_{i} \in \mathbb{R}(\text{or } \mathbb{C})$, the following linear transformation $A : V \to \mathbb{R}(\text{or } \mathbb{C})$ is called a linear form.

$$ A(\mathbf{x}) := \sum\limits_{i=1}^{n} a_{i}x_{i} $$

In this case, $\mathbf{x} = \begin{bmatrix} x_{1} & \cdots & x_{n} \end{bmatrix}^{T}$.

Generalization

For a given inner product space $(V, \left< \cdot, \cdot \right>)$ and $\mathbf{a} \in V$, the following linear functional $A : V \to \mathbb{F}$ is called a linear form.

$$ A(\mathbf{x}) = \left< \mathbf{a}, \mathbf{x} \right> $$

Here, $\mathbb{F}$ is a field of the vector space $V$.

Matrix Form

If $a_{i}$, $x_{i}$ are real numbers, then it is called a linear form on space $\mathbb{R}^{n}$. Moreover, if constants and variables are represented as column vectors like $\mathbf{a}=\begin{bmatrix} a_{1} & \cdots & a_{n} \end{bmatrix}^{T}$, $\mathbf{x}=\begin{bmatrix} x_{1} & \cdots & x_{n} \end{bmatrix}^{T}$, the linear form can be expressed as a matrix inner product.

$$ \mathbf{a} \cdot \mathbf{x} = \mathbf{a}^{T} \mathbf{x} = \begin{bmatrix} a_{1} & \cdots & a_{n} \end{bmatrix} \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix} =\sum \limits _{i=1}^{n} a_{i}x_{i} $$