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 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] = [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].

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

Examples

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,…,x_n). The subspace, identified with R^m, consists of all n-tuples such that the last n-m entries are zero: (x₁,…,x_m,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 n−m coordinates. The quotient space Rⁿ/ 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,

$V=U\oplus W$

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

$0\to U\to V\to V/U\to 0.\,$

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 (Halmos 1974, Theorem 22.2):

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

Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x ∈ 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

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

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] 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

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

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).

See also

• quotient set
• quotient group
• quotient module
• quotient space (topology)

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.