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 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 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] = [αx] for all α ∈ K, and
• [x] + [y] = [x + y].

Also, dim(V /W) = dim V − dim W.

We will construct an explicit basis for V/W. Let (w₁, . . . , w_n) be a basis of W, and extend it to a basis (w₁, . . . , w_n, v₁, . . . , v_k) of V. Note that with this notation, dim W = n and dim V = n + k. We claim that (v₁ + W, . . . , v_k + W) forms a basis of V /W, in other words, ([v₁], [v₂], . . . , [v_k]).

If so, we will then have dim(V /W) = k = (n + k) − n = dim V − dim W as claimed.