In Terms of Determinants
Over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant.
Over a commutative ring R, one must be slightly more careful: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. Therefore GL(n, R) may be defined as the group of matrices whose determinants are units.
Over a non-commutative ring R, determinants are not at all well behaved. In this case, GL(n, R) may be defined as the unit group of the matrix ring M(n, R).
Read more about this topic: General Linear Group
Famous quotes containing the word terms:
“It is not [the toddler’s] job yet to consider other people’s feelings, he has to come to terms with his own first. If he hits you and you hit him back to “show him what it feels like,” you will have given a lesson he is not ready to learn. He will wail as if hitting was a totally new idea to him. He makes no connections between what he did to you and what you then did to him; between your feelings and his own.”
—Penelope Leach (20th century)