Quotient space (linear algebra)

Short description: Vector space consisting of affine subsets

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

Definition

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

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

[math]\displaystyle{ [x] = x + N }[/math]

since it is given by

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

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

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

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 [math]\displaystyle{ V/N }[/math] into a vector space over [math]\displaystyle{ \mathbb{K} }[/math] with [math]\displaystyle{ N }[/math] being the zero class, [math]\displaystyle{ [0] }[/math].

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

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

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, ..., 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 equivalence 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.

Polynomial Vector Space

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

General Subspaces

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

[math]\displaystyle{ V=U\oplus W }[/math]

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

Lebesgue Integrals

An important example of a functional quotient space is an 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

[math]\displaystyle{ 0\to U\to V\to V/U\to 0.\, }[/math]

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:

[math]\displaystyle{ \mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U). }[/math]

Let T : V → W 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 : 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

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

When X is complete, then the quotient space X/M is complete with respect to the norm, and therefore 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. 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 on X/M by

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

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.

See also

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

Sources

• Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-3-319-11079-0. 
• Dieudonné, Jean (1976), Treatise on Analysis, 2, Academic Press, ISBN 978-0122155024 
• Halmos, Paul Richard (1974). Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-90093-4. 
• Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9. 
• Roman, Steven (2005). Advanced Linear Algebra. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-24766-1.