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linear dependence[′lin·ē·ər di′pen·dəns]
(mathematics), a relationship of the form
(*) C1u1 + C2u2 + ... + Cnun = 0
where C1, C2,..., Cn are numbers, at least one of which is not zero, and u1, u2,..., un are various mathematical objects for which the operations of addition and multiplication by a number are defined. The dependence relation (*) is said to be linear in the objects u1, u2,..., un, since each of them enters (*) linearly, that is, the degree of each ui in (*) is one. The equality sign in the above relation may have various meanings and its sense must be specified in each particular case.
The concept of linear dependence is used in many branches of mathematics. We may thus speak, for example, of linear dependence between vectors, between functions of one or several variables, and between elements of a vector space. If the objects u1, u2,..., un are connected by a relation of linear dependence, then we say that they are linearly dependent. In the opposite case we say that they are linearly independent. If the objects u1, u2,..., un are linearly dependent, then at least one of them is a linear combination of the others, that is,
ui = α1u1 + ... + αi - 1ui-1 + αi + 1ui + 1 + ... + αnun
Continuous functions of one variable
u1 = ϕ1(x), u2 = ϕ2(x), . . ., un = ϕn (x)
are said to be linearly dependent if they are connected by a relation of the form (*), in which the equality sign is understood as an identity in x. In order for the functions Φ1(x), Φ2(x), . . ., Φn(x) defined on an interval a ≤ x ≤ b to be linearly dependent, it is necessary and sufficient that their Gramian vanish, the Gramian in this case being the determinant
However, if the functions Φ1(x), Φ2(x),..., Φn(x) are solutions of a linear differential equation, then they are linearly dependent if and only if their Wronskian vanishes at at least one point.
Linear forms in M variables
ui = ai 1x1 + ai 2x2 + ... + aimxmi = 1, 2,..., n
are said to be linearly dependent if they are connected by a relation of the form (*), in which the equality sign is understood as an identity in all the variables x1, x2,..., xm. In order for n linear forms in n variables to be linearly dependent, it is necessary and sufficient that the following determinant vanish: