The problem of cutting subspaces

This paper treats a natural and simply formulated problem in multidimensional geometry and linear algebra that has apparently escaped the attention of investigators. The author arrived at this problem in reflecting over a question posed by L. D. Ivanov. The greater attention is given to the geometric case of a real linear space. In this case the results are far from definitive, in contrast to the case of a linear space over the field of complex numbers.