Quotient space (linear algebra)

This article is about quotients of vector spaces. For quotients of topological spaces, see Quotient space (topology).

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

Definition

Formally, the construction is as follows. ^[1] Let ${\displaystyle V}$ be a vector space over a field ${\displaystyle \mathbb {K} }$, and let ${\displaystyle N}$ be a subspace of ${\displaystyle V}$. We define an equivalence relation ${\displaystyle \sim }$ on ${\displaystyle V}$ by stating that ${\displaystyle x\sim y}$ if ${\displaystyle x-y\in N}$. That is, ${\displaystyle x}$ is related to ${\displaystyle y}$ if and only if one can be obtained from the other by adding an element of ${\displaystyle N}$. This definition implies that any element of ${\displaystyle N}$ is related to the zero vector; more precisely, all the vectors in ${\displaystyle N}$ get mapped into the equivalence class of the zero vector.

The equivalence class – or, in this case, the coset – of ${\displaystyle x}$ is defined as

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

and is often denoted using the shorthand ${\displaystyle [x]=x+N}$.

The quotient space ${\displaystyle V/N}$ is then defined as ${\displaystyle V/_{\sim }}$, the set of all equivalence classes induced by ${\displaystyle \sim }$ on ${\displaystyle V}$. Scalar multiplication and addition are defined on the equivalence classes by ^{[2] [3]}

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

It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives). These operations turn the quotient space ${\displaystyle V/N}$ into a vector space over ${\displaystyle \mathbb {K} }$ with ${\displaystyle N}$ being the zero class, ${\displaystyle [0]}$.

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

Alternatively phrased, the quotient space ${\displaystyle V/N}$ is the set of all affine subsets of ${\displaystyle V}$ which are parallel to ${\displaystyle N}$. ^[4]

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

General Subspaces

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

${\displaystyle V=U\oplus W}$

then the quotient space V/U is naturally isomorphic to W. ^[5]

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

${\displaystyle 0\to U\to V\to V/U\to 0.\,}$

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: ^{[6] [7]}

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

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

${\displaystyle \|[x]\|_{X/M}=\inf _{m\in M}\|x-m\|_{X}=\inf _{m\in M}\|x+m\|_{X}=\inf _{y\in [x]}\|y\|_{X}.}$

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. ^[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 on X/M by

${\displaystyle q_{\alpha }([x])=\inf _{v\in [x]}p_{\alpha }(v).}$

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. ^[9]

See also

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

References

1. Halmos (1974) pp. 33-34 §§ 21-22
2. Katznelson & Katznelson (2008) p. 9 § 1.2.4
3. Roman (2005) p. 75-76, ch. 3
4. Axler (2015) p. 95, § 3.83
5. Halmos (1974) p. 34, § 22, Theorem 1
6. Axler (2015) p. 97, § 3.89
7. Halmos (1974) p. 34, § 22, Theorem 2
8. Dieudonné (1976) p. 65, § 12.14.8
9. Dieudonné (1976) p. 54, § 12.11.3

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 , vol. 2, Academic Press, ISBN   978-0122155024
• Halmos, Paul Richard (1974) [1958]. 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.