Quotient space (linear algebra)

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 $$\mathbb{K}$$, and let $$N$$ be a subspace of $$V$$. We define an equivalence relation $$\sim$$ on $$V$$ by stating that $$x \sim y$$ if $x - y \in N$. 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
 * $$[x] = \{ x + n: n \in N \}$$

The quotient space $$V/N$$ is then defined as $$V/_\sim$$, the set of all equivalence classes induced by $$\sim$$ on $$V$$. Scalar multiplication and addition are defined on the equivalence classes by 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 $$\mathbb{K}$$ with $$N$$ being the zero class, $$[0]$$.
 * $$\alpha [x] = [\alpha x]$$ for all $$\alpha \in \mathbb{K}$$, and
 * $$[x] + [y] = [x+y]$$.

The mapping that associates to $v \in V$ the equivalence class $$[v]$$ is known as the quotient map.

Alternatively phrased, the quotient space $$V/N$$ is the set of all affine subsets of $$V$$ which are parallel to $N$.

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.)

Subspaces of Cartesian Space
Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1, ..., xn). The subspace, identified with Rm, consists of all n-tuples such that the last n − m entries are zero: (x1, ..., xm, 0, 0, ..., 0). Two vectors of Rn are in the same equivalence class modulo the subspace if and only if they are identical in the last n − m coordinates. The quotient space Rn/Rm is isomorphic to  Rn−m in an obvious manner.

Polynomial Vector Space
Let $$\mathcal{P}_3(\mathbb{R})$$ be the vector space of all cubic polynomials over the real numbers. Then $$\mathcal{P}_3(\mathbb{R}) / \langle x^2 \rangle $$ is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only. For example, one element of the quotient space is $$\{x^3 + a x^2 - 2x + 3 : a \in \mathbb{R}\}$$, while another element of the quotient space is $$\{a x^2 + 2.7 x : a \in \mathbb{R}\}$$.

General Subspaces
More generally, if V is an (internal) direct sum of subspaces U and W,
 * $$V=U\oplus W$$

then the quotient space V/U is naturally isomorphic to W.

Lebesgue Integrals
An important example of a functional quotient space is an Lp space.

Properties
There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence
 * $$0\to U\to V\to V/U\to 0.\,$$

If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U:
 * $$\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).$$

Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x in V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).

The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).

Quotient of a Banach space by a subspace
If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M by
 * $$ \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X = \inf_{m \in M} \|x+m\|_X = \inf_{y\in [x]}\|y\|_X. $$

Examples
Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f &isin; C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R.

If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.

Generalization to locally convex spaces
The quotient of a locally convex space by a closed subspace is again locally convex. Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α &isin; A} where A is an index set. Let M be a closed subspace, and define seminorms qα on X/M by


 * $$q_\alpha([x]) = \inf_{v\in [x]} p_\alpha(v).$$

Then X/M is a locally convex space, and the topology on it is the quotient topology.

If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M.