# Linear Form

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## 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 *x*_{1}, *x*_{2}, ..., *x _{n}* is given by the equality

*f*(*x*_{1}, *x*_{2}, ⋯, *x _{n}*) =

*a*

_{1}

*x*

_{1}+

*a*

_{2}

*x*

_{2}+ ⋯ +

*a*

_{n}x_{n}where *a*_{1}, *a*_{2}, ..., *a _{n}* are constants. If we interpret

*x*

_{1},

*x*

_{2}, ...,

*x*as the coordinates of a vector x in an

_{n}*n*-dimensional vector space, then

*f*will satisfy the relation

*f*(αx + βy) = α*f*(x) + β*f*(y)

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