| Using predicate logic, the concept of a linear problem is formalized. The class of linear problems is huge, diverse, complex, and important. Linear and randomized linear algorithms are formalized. For each linear problem, a linear algorithm is constructed that solves the problem and a randomized linear algorithm is constructed that completely solves it, that is, for any data of the problem, the output set of the randomized linear algorithm is identical to the solution set of the problem. We obtain a single machine, referred to as the Universal (Randomized) Linear Machine, which (completely) solves every instance of every linear problem. Conversely, for each randomized linear algorithm, a linear problem is constructed that the algorithm completely solves. These constructions establish a one-to-one and onto correspondence from equivalence classes of linear problems to equivalence classes of randomized linear algorithms. Our construction of (randomized) linear algorithms to (completely) solve linear problems as well as the algorithms themselves are based on Fourier Elimination and have superexponential complexity. However, there is no evidence that the inefficiency of our methods is unavoidable relative to the difficulty of the problem. |