In this paper, we propose a novel reduced-rank adaptive filtering algorithm based on set-theoretic adaptive filtering. We discuss the orthonormality of the transformation (rank-reduction) matrix. We present, under the assumption that the transformation matrix has an orthonormal structure, an interpretation of the proposed algorithm in the original (fullsize) vector space. The interpretation suggests that the use of an orthonormal transformation matrix leads to performance depending solely on the subspace spanned by the column vectors of the matrix but not on the matrix itself. This is verified by simulations, and the numerical examples demonstrate the efficacy of the proposed algorithm.