# 33.4 Quotient Spaces

Let V be a vector space, and U a subspace of V. The set { v + U; v in V } is again a vector space, the quotient space (or factor space) of V modulo U.

By definition of row spaces, a quotient space is not a row space. (One reason to describe quotient spaces here is that for general vector spaces at the moment no factor structures are supported.)

Quotient spaces in GAP are formed from two spaces using the `/` operator. See the sections Operations for Quotient Spaces and Functions for Quotient Spaces for an overview of applicable operators and functions, and Quotient Space Records for details of the implementation.

Bases for Quotient Spaces of Row Spaces

A basis B of a quotient V / U for row spaces V and U is best described by bases of V and U. If B is a basis without special properties then it will delegate the work to a semi-echelonized basis. The concept of semi-echelonized bases makes sense also for quotient spaces of row spaces since for any semi-echelonized basis of U the set S of pivot columns is a subset of the set of pivot columns of a semi-echelonized basis of V. So the cosets v + U for basis vectors v with pivot column not in S form a semi-echelonized basis of V / U. The canonical basis of V / U is the semi-echelonized basis derived in that way from the canonical basis of V (see CanonicalBasis).

See Functions for Quotient Spaces for details about the bases.

GAP 3.4.4
April 1997