The generation of a two-dimensional linear vector space in a coherent optical data processing system and its specific application to the gradient operator is discussed using theoretical and experimental results. The vector space is created by superposing the outputs of two Fourier optical systems having light of mutually orthogonal polarizations. As a result, the total amplitude of the signal in the output plane is a vector sum of the signals from the systems. If each one of the systems performs one of the partial derivative operations of the transverse gradient, and if the inputs to both systems are identical, then the output is the vector sum of the partial derivatives or the transverse gradient operation. The experimental program is heavily oriented toward the realization of the optimum approximation to the jomega(x)î and jomega(y)j; filters, rather than using various binary-type filters. Problems with film and lens noise are also discussed.