# 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 (Halmos 1974, §21-22). Let *V* be a vector space over a field *K*, and let *N* be a subspace of *V*. We define an equivalence relation ~ on *V* by stating that *x* ~ *y* if *x* − *y* ∈ *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*∈*N*}.

The quotient space *V*/*N* is then defined as *V*/~, the set of all equivalence classes over *V* by ~. Scalar multiplication and addition are defined on the equivalence classes by

- α[
*x*] = [α*x*] for all α ∈*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 representative). These operations turn the quotient space *V*/*N* into a vector space over *K* with *N* being the zero class, [0].

The mapping that associates to *v* ∈ *V* the equivalence class [*v*] is known as the **quotient map**.

## Examples

Let *X* = **R**^{2} 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*. This gives one way in which to visualize quotient spaces geometrically.

Another example is the quotient of **R**^{n} by the subspace spanned by the first *m* standard basis vectors. The space **R**^{n} consists of all *n*-tuples of real numbers (*x*_{1},…,*x*_{n}). The subspace, identified with **R**^{m}, consists of all *n*-tuples such that the last *n-m* entries are zero: (*x*_{1},…,*x*_{m},0,0,…,0). Two vectors of **R**^{n} are in the same congruence class modulo the subspace if and only if they are identical in the last *n*−*m* coordinates. The quotient space **R**^{n}/ **R**^{m} is isomorphic to **R**^{n−m} in an obvious manner.

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

then the quotient space *V*/*U* is naturally isomorphic to *W* (Halmos 1974, Theorem 22.1).

An important example of a functional quotient space is a L^{p} 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

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* (Halmos 1974, Theorem 22.2):

Let *T* : *V* → *W* be a linear operator. The kernel of *T*, denoted ker(*T*), is the set of all *x* ∈ *V* such that *Tx* = 0. The kernel is a subspace of *V*. The first isomorphism theorem of linear algebra 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

The quotient space *X*/*M* is complete with respect to the norm, so it is a Banach space.

### 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* ∈ *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 (Dieudonné 1970, 12.14.8). Indeed, suppose that *X* is locally convex so that the topology on *X* is generated by a family of seminorms {*p*_{α} | α ∈ *A*} where *A* is an index set. Let *M* be a closed subspace, and define seminorms *q*_{α} by on *X*/*M*

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* (Dieudonné 1970, 12.11.3).

## References

- Halmos, Paul (1974),
*Finite dimensional vector spaces*, Springer, ISBN 978-0-387-90093-3. - Dieudonné, Jean (1970),
*Treatise on analysis, Volume II*, Academic Press.