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linear form[′lin·ē·ər ′fȯrm]
a form of the first degree. A linear form in n variables x1, x2, ..., xn is given by the equality
f(x1, x2, ⋯, xn) = a1x1 + a2x2 + ⋯ + anxn
where a1, a2, ..., an are constants. If we interpret x1, x2, ..., xn 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.