What is Quotient space (linear algebra)?
'''Quotient space''' is a term from [[linear algebra]]. It is a [[vector space]] that is made using a [[vector subspace]], and a [[parallel projection]]. The elements of quotient spaces are [[Equivalence class|equivalence classes]]. A simple example of a quotient space is the following: take the Cartesian coordinate system and draw a straight line which goes through the origin of that system. All the lines which are parallel to that line are a quotient space. == More detailed description == In [[linear algebra]], the '''quotient space''' of a [[vector space]] V by a subspace N, also said V/N, is the set of vectors v and w of V such that v-w is in N. We can also say that v added by any vector of N is in the [[Equivalence relation|class]] of v, also called [v]. The set of v and vectors in V such that v plus all those vectors are in N are written V/N (V/N = {v + N | v ∈ V }). Also, two vectors of V are equivalent if their substraction are in W. The quotient space shares this two propierties: * α[''x''] = [α''x''] for all α ∈ ''K'', and * [''x''] + [''y''] = [''x'' + ''y'']. Also, [[Dimension|dim]](V /W) = dim V − dim W. We will construct an explicit basis for V/W. Let (w<sub>1</sub>, . . . , w<sub>n</sub>) be a [[Basis (linear algebra)|basis]] of W, and extend it to a basis (w<sub>1</sub>, . . . , w<sub>n</sub>, v<sub>1</sub>, . . . , v<sub>k</sub>) of V. Note that with this notation, dim W = n and dim V = n + k. We claim that (v<sub>1</sub> + W, . . . , v<sub>k</sub> + W) forms a basis of V /W, in other words, ([v<sub>1</sub>], [v<sub>2</sub>], . . . , [v<sub>k</sub>]). If so, we will then have dim(V /W) = k = (n + k) − n = dim V − dim W as claimed. [[Category:Linear algebra]]