Linear Functional

linear-algebra

Definition

Def

Let $V$ be a vector space over a field $\mathbb{F}$. A linear functional (or linear form) is a linear map $f:V\to \mathbb{F}$ satisfying:

1. Additivity: $f(u+v)=f(u)+f(v)$
2. Homogeneity: $f(\alpha v)=\alpha f(v)$

for all $u,v\in V$ and $\alpha \in \mathbb{F}$. Equivalently:

$$f(\alpha u+\beta v)=\alpha f(u)+\beta f(v)$$