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The concept of linearity has two applications in mathematics.
(1) In a mathematical structure which has the concept of addition incorporated into it, a linear function f is one for which f(a + b) = f(a) + f(b); in some uses of the term, mainly involving vectors, it must also be true that f(a × v) = a × f(v), where a is a scalar multiple (something to make the vector v longer or shorter but not change its direction). Essentially, then, a linear function is one for which it does not matter if addition is calculated before or after the function itself is calculated.
(2) A linear equation is one in which all the terms are raised to only the power 1, such as ax + by = c (which is the general equation for the straight line in Cartesian co-ordinates, hence the name). It is usually much easier to solve linear equations than non-linear ones. This is particularly true of linear differential equations, which can often be exactly solved, whereas non-linear differential equations usually cannot. A linear differential equation will look like af(x) = bf\'(x), for example (where f\' is the first derivative of the function f), while a non-linear equation might take the form af(x)f\'\'(x) + bf\'(x) = c (where f\'\' is the second derivative of the first derivative of f(x)). Many physical systems are governed by the behaviour of non-linear differential equations (turbulence in liquids, for example), and in the past it has been usual to try to understand these systems by approximating the correct differential equation with a linear differential equation which can be exactly solved. Today, with modern advances in computer technology, it has become possible to find approximate solutions to the correct differential equation, which has led to new understandings of many physical theories. SMcL
See also chaos theory. |
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