In this paper a new algebraic representation of linear time-variant dynamic systems is developed. It is shown that Walsh functions can be used to provide such a representation up to any desired precision. Due to the orthogonality of the Walsh functions, the required precision only depends on the number of Walsh functions used in the underlying Walsh - Fourier analysis. The resulting linear algebraic model bears some resemblance to the well-known Laplace transform and especially in the case of linear time-invariant systems there is even a direct link between the two descriptions. Based upon this result new procedures for simulation, system identification and controller design can be obtained. This is illustrated by calculating stairstep approximations of the inverse Laplace transform of rational and irrational systems as well as the design of a time-variant multivariable PI controller for a sixth-order linear time-variant system.