The purpose of the manual is to acquaint the reader with some basic ideas of modern mathematics that have mechanical or physical motivations. Its feature, reflected in the title, is the purposeful use of geometric, visual images. The author pays a lot of attention to the analysis of specific examples. Addressed to students of the third or fourth years of undergraduate studies in the specialty "Mathematics." Readers are required to master the material of the usual program of the first two courses of mathematics faculties.