# 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

• $\alpha [x]=[\alpha x]$  for all $\alpha \in \mathbb {K}$ , 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 $\mathbb {K}$  with $N$  being the zero class, $$ .

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 $N$ .

## 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 ${\mathcal {P}}_{3}(\mathbb {R} )$  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 $\{x^{3}+ax^{2}-2x+3:a\in \mathbb {R} \}$ , while another element of the quotient space is $\{ax^{2}+2.7x:a\in \mathbb {R} \}$ .

### 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 : 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

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

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

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

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.