In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read "V mod N" or "V by N"). Definition Formally, the construction is as follows. Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if . That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector; more precisely, all the vectors in N get mapped into the equivalence class of the zero vector. The equivalence class – or, in this case, the coset – of x is often denoted [x] = x + N since it is given by [x] = {x + n : n ∈ N}. The quotient space V/N is then defined as V/~, the set of all equivalence classes induced by ~ on V. Scalar multiplication and addition are defined on the equivalence classes by α[x] = [αx] for all α ∈ K, and [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/N into a vector space over K with N being the zero class, [0]. The mapping that associates to the equivalence class [v] is known as the quotient map. Alternatively phrased, the quotient space is the set of all affine subsets of which are parallel to Examples Lines in Cartesian Plane Let 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