# Quotient space (linear algebra)

Short description: Vector space consisting of affine subsets

In linear algebra, the quotient of a vector space $\displaystyle{ V }$ by a subspace $\displaystyle{ N }$ is a vector space obtained by "collapsing" $\displaystyle{ N }$ to zero. The space obtained is called a quotient space and is denoted $\displaystyle{ V/N }$ (read "$\displaystyle{ V }$ mod $\displaystyle{ N }$" or "$\displaystyle{ V }$ by $\displaystyle{ N }$").

## Definition

Formally, the construction is as follows. Let $\displaystyle{ V }$ be a vector space over a field $\displaystyle{ \mathbb{K} }$, and let $\displaystyle{ N }$ be a subspace of $\displaystyle{ V }$. We define an equivalence relation $\displaystyle{ \sim }$ on $\displaystyle{ V }$ by stating that $\displaystyle{ x \sim y }$ if $\displaystyle{ x - y \in N }$. That is, $\displaystyle{ x }$ is related to $\displaystyle{ y }$ if one can be obtained from the other by adding an element of $\displaystyle{ N }$. From this definition, one can deduce that any element of $\displaystyle{ N }$ is related to the zero vector; more precisely, all the vectors in $\displaystyle{ N }$ get mapped into the equivalence class of the zero vector.

The equivalence class – or, in this case, the coset – of $\displaystyle{ x }$ is often denoted

$\displaystyle{ [x] = x + N }$

since it is given by

$\displaystyle{ [x] = \{ x + n: n \in N \} }$

The quotient space $\displaystyle{ V/N }$ is then defined as $\displaystyle{ V/_\sim }$, the set of all equivalence classes induced by $\displaystyle{ \sim }$ on $\displaystyle{ V }$. Scalar multiplication and addition are defined on the equivalence classes by

• $\displaystyle{ \alpha [x] = [\alpha x] }$ for all $\displaystyle{ \alpha \in \mathbb{K} }$, and
• $\displaystyle{ [x] + [y] = [x+y] }$.

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 $\displaystyle{ V/N }$ into a vector space over $\displaystyle{ \mathbb{K} }$ with $\displaystyle{ N }$ being the zero class, $\displaystyle{  }$.

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

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

## 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.)

### 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 nm 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 nm coordinates. The quotient space Rn/Rm is isomorphic to Rnm in an obvious manner.

### Polynomial Vector Space

Let $\displaystyle{ \mathcal{P}_3(\mathbb{R}) }$ be the vector space of all cubic polynomials over the real numbers. Then $\displaystyle{ \mathcal{P}_3(\mathbb{R}) / \langle x^2 \rangle }$ 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 $\displaystyle{ \{x^3 + a x^2 - 2x + 3 : a \in \mathbb{R}\} }$, while another element of the quotient space is $\displaystyle{ \{a x^2 + 2.7 x : a \in \mathbb{R}\} }$.

### General Subspaces

More generally, if V is an (internal) direct sum of subspaces U and W,

$\displaystyle{ V=U\oplus W }$

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

$\displaystyle{ 0\to U\to V\to V/U\to 0.\, }$

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:

$\displaystyle{ \mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U). }$

Let T : VW 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 : VW 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

$\displaystyle{ \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X = \inf_{m \in M} \|x+m\|_X = \inf_{y\in [x]}\|y\|_X. }$

When X is complete, then the quotient space X/M is complete with respect to the norm, and therefore a Banach space.[citation needed]

### 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 fC[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 ∈ A} where A is an index set. Let M be a closed subspace, and define seminorms qα on X/M by

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

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.