Quotient space (linear algebra)

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

Definition

Formally, the construction is as follows.<ref>Template:Harvard citation text pp. 33-34 §§ 21-22</ref> Let <math>V</math> be a vector space over a field <math>\mathbb{K}</math>, and let <math>N</math> be a subspace of <math>V</math>. We define an equivalence relation <math>\sim</math> on <math>V</math> by stating that <math>x \sim y</math> iff Template:Nowrap. That is, <math>x</math> is related to <math>y</math> if and only if one can be obtained from the other by adding an element of <math>N</math>. This definition implies that any element of <math>N</math> is related to the zero vector; more precisely, all the vectors in <math>N</math> get mapped into the equivalence class of the zero vector.

The equivalence class – or, in this case, the coset – of <math>x</math> is defined as

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

and is often denoted using the shorthand <math>[x] = x + N</math>.

The quotient space <math>V/N</math> is then defined as <math>V/_\sim</math>, the set of all equivalence classes induced by <math>\sim</math> on <math>V</math>. Scalar multiplication and addition are defined on the equivalence classes by<ref>Template:Harvard citation text p. 9 § 1.2.4</ref><ref>Template:Harvard citation text p. 75-76, ch. 3</ref>

• <math>\alpha [x] = [\alpha x]</math> for all <math>\alpha \in \mathbb{K}</math>, and
• <math>[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>V/N</math> into a vector space over <math>\mathbb{K}</math> with <math>N</math> being the zero class, <math>[0]</math>.

The mapping that associates to Template:Nowrap the equivalence class <math>[v]</math> is known as the quotient map.

Alternatively phrased, the quotient space <math>V/N</math> is the set of all affine subsets of <math>V</math> which are parallel to Template:Nowrap<ref>Template:Harvard citation text p. 95, § 3.83</ref>

Examples

Lines in Cartesian Plane

Let Template:Nowrap 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 Template:Nowrap. The subspace, identified with R^m, consists of all n-tuples such that the last n − m entries are zero: Template:Nowrap. 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>\mathcal{P}_3(\mathbb{R})</math> be the vector space of all cubic polynomials over the real numbers. Then <math>\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>\{x^3 + a x^2 - 2x + 3 : a \in \mathbb{R}\}</math>, while another element of the quotient space is <math>\{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>V=U\oplus W</math>

then the quotient space V/U is naturally isomorphic to W.<ref>Template:Harvard citation text p. 34, § 22, Theorem 1</ref>

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>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:<ref>Template:Harvard citation text p. 97, § 3.89</ref><ref>Template:Harvard citation text p. 34, § 22, Theorem 2</ref>

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

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 Template:Nowrap 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.<ref>Template:Harvard citation text p. 65, § 12.14.8</ref> 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>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.<ref>Template:Harvard citation text p. 54, § 12.11.3</ref>

See also

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

References

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Sources

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