Linear Dependence

linear dependence

[′lin·ē·ər di′pen·dəns] (mathematics) The property of a set of vectorsv, …,v_nin a vector space for which there exists a linear combination such that a₁v₁+ ··· + a_nv_n= 0, and at least one of the scalars a_iis not zero.

Linear Dependence

(mathematics), a relationship of the form

(*) C₁u₁ + C₂u₂ + ... + C_nu_n = 0

where C₁, C₂,..., C_n are numbers, at least one of which is not zero, and u₁, u₂,..., u_n 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,..., u_n, since each of them enters (*) linearly, that is, the degree of each u_i 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,..., u_n 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,..., u_n are linearly dependent, then at least one of them is a linear combination of the others, that is,

u_i = α₁u₁ + ... + α_{i - 1}u_i-1 + α_{i + 1}u_{i + 1} + ... + α_nu_n

Continuous functions of one variable

u₁ = ϕ₁(x), u₂ = ϕ₂(x), . . ., u_n = ϕ_n (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), . . ., Φ_n(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

Where

However, if the functions Φ₁(x), Φ₂(x),..., Φ_n(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

u_i = a_{i 1}x₁ + a_{i 2}x₂ + ... + a_imx_mi = 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,..., x_m. In order for n linear forms in n variables to be linearly dependent, it is necessary and sufficient that the following determinant vanish:

单词	linear dependence
释义	linear dependence linear dependence n. The property of a set of vectors having at least one linear combination equal to zero when at least one of the coefficients is not equal to zero. Linear Dependence linear dependence [′lin·ē·ər di′pen·dəns] (mathematics) The property of a set of vectorsv, …,v_nin a vector space for which there exists a linear combination such that a₁v₁+ ··· + a_nv_n= 0, and at least one of the scalars a_iis not zero. Linear Dependence (mathematics), a relationship of the form () C₁u₁ + C₂u₂ + ... + C_nu_n = 0 where C₁, C₂,..., C_n are numbers, at least one of which is not zero, and u₁, u₂,..., u_n 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,..., u_n, since each of them enters () linearly, that is, the degree of each u_i* 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,..., u_n 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,..., u_n are linearly dependent, then at least one of them is a linear combination of the others, that is, u_i = α₁u₁ + ... + α_{i - 1}u_i-1 + α_{i + 1}u_{i + 1} + ... + α_nu_n Continuous functions of one variable u₁ = ϕ₁(x), u₂ = ϕ₂(x), . . ., u_n = ϕ_n (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), . . ., Φ_n(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 Where However, if the functions Φ₁(x), Φ₂(x),..., Φ_n(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 u_i = a_{i 1}x₁ + a_{i 2}x₂ + ... + a_imx_mi = 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,..., x_m. In order for n linear forms in n variables to be linearly dependent, it is necessary and sufficient that the following determinant vanish:
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