An Etymological Dictionary of Astronomy and Astrophysics
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An equation in which the → dependent variable y and all its differential coefficients occur only in the first degree. A linear differential equation of → order  order n has the form:
f_n(x)y⁽ⁿ⁾ + f_n-1(x)y^(n-1) + ... + f₁(x)y^' + f₀(x)y = Q(x),
where f₀(x), f₁(x), ..., f_n(x) and Q(x) are each continuous functions of x defined on a common interval I and f_n(x)≠ 0 in I. A linear differential equation cannot have, for example, terms such as y² or (y^')^1/2. See also: → homogeneous linear differential equation; → nonhomogeneous linear differential equation.

→ linear; → differential; → equation.

A system of three charges +q, -2q, and +q, arranged along a line to form an axial quadrupole. The → electric potential  V due to a linear quadrupole varies as 1/r³, whereas the → electric intensity  E varies as 1/r⁴.

→ linear; → electric; → quadrupole.

An equation composed of first degree variables and representing a straight line.

→ linear; → equation.

A function expressed by a → first degree equation that can be graphically represented in the → Cartesian coordinate plane by a → straight line.

→ linear; → function.

An instability that can be described (to first-order accuracy) by linear (or tangent linear) equations.

→ linear; → instability.

The product of an object's → mass and → velocity. It is a → vector and points in the same direction as the velocity vector. Linear momentum is distinguished from → angular momentum. When there is no opportunity for confusion, usually the term momentum is used instead of linear momentum.

→ linear; → momentum.

Assumption that the variations in the plasma parameters, due to the presence of waves, are small (to the first order) as compared to the undisturbed parameters. This makes it possible to linearize equations by dropping out second order (and higher) nonlinear terms.

→ linear; → perturbation; → theory.

Of an electromagnetic radiation, a → polarization in which the electric vibrations are confined to one plane along the direction of propagation. Also called → plane polarization. See also → circular polarization.

→ linear; → polarization.

A procedure for finding the maximum or minimum of a → linear function where the → arguments are subject to linear → constraints. For problems involving more than two variables or problems involving a large number of constraints, solution methods used are those that are adaptable to computers. A well-known such → algorithm is the → simplex method.

→ linear; → programmings.

In statistics, a regression method that establishes a linear relationship between two random variables.

→ linear; → regression.

The real, physical size, as opposed to angular size.

→ linear; → size.

Physics: A → dynamical system whose evolution is a linear process. If a change in any variable at some initial time produces a change in the same or some other variable at some later time, twice as large a change at the same initial time will produce twice as large a change at the same later time.

→ linear; → system.

The rate of change of the position of an object that is traveling along a straight path. In other words, the velocity of an object when its moving direction is not changing. For a given → angular velocity (ω), the linear velocity v of the particle is directly proportional to the distance of the particle from the center of the circular path: v = ω ×r.

→ linear; → velocity.

The property, condition, or state of being linear.
Math.: A relationship between two variables so that when plotted on a graph they yield a straight line.

→ linear; → -ity.

A process of reduction to linear form by appropriate change of variables or by approximation.

Verbal noun of → linearize.

To make linear; give linear form to.

→ linear + → -ize.

A differential equation that has been derived from an original nonlinear equation.

Linearized, p.p. of → linearize; → differential; → equation.

In a manner characterized by first-degree algebraic terms.

Adverb of → linear.

A set of objects x₁, x₂, ..., x_n (→ vectors, → matrices, → polynomials, etc.) on a given set if there is a linear combination of them: a₁x₁ + a₂x₂ + ... + a_nx_n, which is zero, but at least one of the coefficients is non-zero. For example the binomials (2x + y) and (6x + 3y) are linearly dependent, since 3(2x + y) - (6x + 3y) = 0.

→ linearly; → dependent.

1) A set of objects x₁, x₂, ..., x_n (→ vectors, → matrices, → polynomials, etc.) if it si not → linearly dependent.
2) Of two particular solutions (y₁, y₂) of a → homogeneous linear differential equation of the second order (y^'' + a₁ y^' + a₂y = 0) on an interval [a, b], if their ratio in this interval is not a constant: y₁/y₂≠ constant.

→ linearly; → independent.