Mode of a linear field

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In physics, a vector field is linear if it is a solution of a set of linear equations E. For instance, in physics, the electromagnetic field in vacuum, defined in the usual (3 + 1)-dimensional space S, obeys Maxwell's equations. A linear combination of electromagnetic fields, with constant, real coefficients, is a new field which obeys Maxwell's equations.

The solutions of the linear equations are represented in a real vector space M. A radius of M, which represents proportional solutions, is called a "mode".[clarification needed]

A norm may be defined. For instance, in electromagnetism, it is usually the energy of the solution assuming that there is no other field in S. From the norm are defined the orthogonality and the scalar product of solutions. The orthogonality of solutions extends to the corresponding modes.