Definitions

# 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).

## Definition

Formally, the construction is as follows . 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 − yN. That is, x is related to y if one can be obtained from the other by adding an element of N. The equivalence class of x is often denoted

[x] = x + N
since it is given by
[x] = {x + n : nN}.

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] = [αx] 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].

## Examples and properties

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. This gives one way in which to visualize quotient spaces geometrically.

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 only the first m entries are non-zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence 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.

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

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

If U is a subspace of V, the codimension of U in V is defined to be the dimension of V/U. If V is finite-dimensional, this is just the difference in the dimensions of V and U :

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

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

$0to Uto Vto V/Uto 0.,$

Let T : VW be a linear operator. The kernel of T, denoted ker(T), is the set of all xV 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 : 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

$| \left[x\right] |_\left\{X/M\right\} = inf_\left\{m in M\right\} |x-m|_X.$
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]. 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 q&alpha by on X/M

$q_alpha\left(\left[x\right]\right) = inf_\left\{xin \left[x\right]\right\} p_alpha\left(x\right).$

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 .