A solution theory for switched linear differential-algebraic equations (DAEs) is developed. To allow for non-smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise-smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given.