Linear Form

linear form [′lin·ē·ər ′fȯrm]

(mathematics)

A homogeneous polynomial of the first degree.

---

Linear Form 

a form of the first degree. A linear form in n variables x₁, x₂, ..., x_n is given by the equality

f(x₁, x₂, ⋯, x_n) = a₁x₁ + a₂x₂ + ⋯ + a_nx_n

where a₁, a₂, ..., a_n are constants. If we interpret x₁, x₂, ..., x_n as the coordinates of a vector x in an n-dimensional vector space, then f will satisfy the relation

f(αx + βy) = αf(x) + βf(y)

(where x and y are vectors and α and β are numbers). Conversely, a form satisfying this relation is linear.