Some new facts about linear maps that preserve the orthogonal group

In this article, Albert Wei's result which described the forms of the linear maps that preserve the set of orthogonal matrices is treated and its new proof is provided. Wei's result is simplified by introducing a new description of the singular linear maps acting into the space n by n real matrices that preserve orthogonal matrices. From this description, one can immediately conclude that the range of such singular maps is isometrically isomorphic with the space (Formula presented.). The proof of the Wei's result that is presented here is based on many new facts about linear maps that preserve the orthogonal group of matrices that are of interest independently of issues considered here.