solutions to linear systems comprise a system of functions which act like vectors in the geometric sense. A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience. The previous definition of a linear system is applicable to SISO (single-input single-output) systems. For MIMO (multiple-input multiple-output) systems, input and output signal vectors (, , , ) are considered instead of input and output signals (, , , .) This definition of a linear system is analogous to the definition of a linear differential equation in calculus, and a linear transformation in linear algebra. Examples A simple harmonic oscillator obeys the differential equation: If then is a linear operator. Letting we can rewrite the differential equation as which shows that a simple harmonic oscillator is a linear system. Other examples of linear systems include those described by , , , and any system described by ordinary linear differential equations. Systems described by , , , , , , , and a system with odd-symmetry output consisting of a linear region and a saturation (constant) region, are non-linear because they don't always satisfy the superposition principle. The output versus input graph of a linear system need not be a straight line through the origin. For example, consider a system described by (such as a constant-capacitance capacitor or a constant-inductance inductor). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid, the output is also a sinusoid, and so its output-input plot is an ellipse centered at the origin rather than a straight line passing through the origin. Also, the output of a linear system can contain harmonics (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by . It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form , using product-to-sum trigonometric identities it can be easily shown that the output is , that is, the output doesn't consist only of sinusoids of same frequency as the input (), but instead also of sinusoids of frequencies and ; furthermore, taking the least common multiple of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is , which is different than that of the