In linear algebra, the quotient of a vector space V {\displaystyle V} by a subspace U {\displaystyle U} is a vector space obtained by "collapsing" U {\displaystyle U} to zero. The space obtained is called a quotient space and is denoted V / U {\displaystyle V/U} (read " V {\displaystyle V} mod U {\displaystyle U} " or " V {\displaystyle V} by U {\displaystyle U} "). Alternatively phrased, the quotient space V / U {\displaystyle V/U} is the set of all affine subsets of V {\displaystyle V} which are parallel to U {\displaystyle U} .
Definition Formally, the construction is as follows. Let V {\displaystyle V} be a vector space over a field K {\displaystyle \mathbb {K} } , and let U {\displaystyle U} be a subspace of V {\displaystyle V} . We define an equivalence relation ∼ {\displaystyle \sim } on V {\displaystyle V} by stating that x ∼ y {\displaystyle x\sim y} iff x − y ∈ U {\displaystyle x-y\in U} . That is, x {\displaystyle x} is related to y {\displaystyle y} if and only if one can be obtained from the other by adding an element of U {\displaystyle U} . This definition implies that any element of U {\displaystyle U} is related to the zero vector; more precisely, all the vectors in U {\displaystyle U} get mapped into the equivalence class of the zero vector. The equivalence class – or, in this case, the coset – of v {\displaystyle v} is defined as
[ v ] := { w : v − w ∈ U } {\displaystyle [v]:=\{w:v-w\in U\}} . Equivalently, [ v ] = { v + u : u ∈ U } {\displaystyle [v]=\{v+u:u\in U\}} , so it is often denoted using the shorthand v + U {\displaystyle v+U} . The quotient space V / U {\displaystyle V/U} is then defined as V / ∼ {\displaystyle V/{\mathord {\sim }}} , the set of all equivalence classes induced by ∼ {\displaystyle \sim } on U {\displaystyle U} . Scalar multiplication and addition are defined on the equivalence classes by
α [ x ] = [ α x ] {\displaystyle \alpha [x]=[\alpha x]} for all α ∈ K {\displaystyle \alpha \in \mathbb {K} } , and
[ x ] + [ y ] = [ x + y ] {\displaystyle [x]+[y]=[x+y]} . It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives). These operations turn the quotient space V / U {\displaystyle V/U} into a vector space over K {\displaystyle \mathbb {K} } with U {\displaystyle U} being the zero class, [ 0 ] {\displaystyle [0]} . The mapping that associates to v ∈ V {\displaystyle v\in V} the equivalence class [ v ] {\displaystyle [v]} is known as the quotient map.
Examples
Lines in Cartesian Plane Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to Y. This gives a way to visualize quotient spaces geometrically. (By re-parameterising these lines, the quotient space can more conventionally be represented as the space of all points along a line through the origin that is not parallel to Y. Similarly, the quotient space for R3 by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a plane which only intersects the line at the origin.)
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