The equations describing many important physical phenomena are linear differential equations. Quantum mechanics is a good example - the Schrödinger equation contains no power of the wavefunction or its derivatives greater than the first, and so is a linear differential equation. Similarly Maxwell's equations contain no power of the electric and magnetic fields or their derivatives greater than the first, and so Maxwell's theory of electromagnetism is also linear. One consequence of a linear theory is that the sum of any two independent solutions is also a solution; this leads to the phenomenon of interference in both optics and quantum mechanics. Even when the underlying equations describing the physics are not linear, we often linearise the problem by looking, for example, at small amplitude disturbances. We may be throwing away some important physics in doing this, but within the domain of validity of the linearising procedure we can still describe small amplitude disturbances in such systems.
In this module we shall study the mathematics which underlies and is common to all these different linear systems. The mathematical techniques and ideas are illustrated by drawing on familiar examples from both classical and quantum physics.