Quotient space (linear algebra)

Formally, the construction is as follows.{{Harvard citation text|Halmos|1974}} pp. 33-34 §§ 21-22 Let $V$ be a vector space over a field $\backslash mathbb\{K\}$, and let $N$ be a subspace of $V$. We define an equivalence relation $\backslash sim$ on $V$ by stating that $x\; \backslash sim\; y$ iff {{nowrap| $x\; -\; y\; \backslash in\; N$}}. That is, $x$ is related to $y$ if and only if one can be obtained from the other by adding an element of $N$. This definition implies 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 – or, in this case, the coset – of $x$ is defined as

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

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

The quotient space $V/N$ is then defined as $V/\_\backslash sim$, the set of all equivalence classes induced by $\backslash sim$ on $V$. Scalar multiplication and addition are defined on the equivalence classes by{{Harvard citation text|Katznelson|Katznelson|2008}} p. 9 § 1.2.4{{Harvard citation text|Roman|2005}} p. 75-76, ch. 3

• $\backslash alpha\; [x]\; =\; [\backslash alpha\; x]$ for all $\backslash alpha\; \backslash in\; \backslash mathbb\{K\}$, 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 representatives). These operations turn the quotient space $V/N$ into a vector space over $\backslash mathbb\{K\}$ with $N$ being the zero class, $[0]$.

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

Alternatively phrased, the quotient space $V/N$ is the set of all affine subsets of $V$ which are parallel to {{nowrap|$N$.}}{{Harvard citation text|Axler|2015}} p. 95, § 3.83

Let {{nowrap|1=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.)

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 {{nowrap|(x₁, ..., x_n)}}. The subspace, identified with R^m, consists of all n-tuples such that the last n − m entries are zero: {{nowrap|(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.

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

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

:$V=U\backslash oplus\; W$

then the quotient space V/U is naturally isomorphic to W.{{Harvard citation text|Halmos|1974}} p. 34, § 22, Theorem 1

An important example of a functional quotient space is an L^p space.

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

:$0\backslash to\; U\backslash to\; V\backslash to\; V/U\backslash to\; 0.\backslash ,$

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:{{Harvard citation text|Axler|2015}} p. 97, § 3.89{{Harvard citation text|Halmos|1974}} p. 34, § 22, Theorem 2

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

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).

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

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

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 {{nowrap|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.

The quotient of a locally convex space by a closed subspace is again locally convex.{{Harvard citation text|Dieudonné|1976}} p. 65, § 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 on X/M by

:$q\_\backslash alpha([x])\; =\; \backslash inf\_\{v\backslash in\; [x]\}\; p\_\backslash alpha(v).$

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.{{Harvard citation text|Dieudonné|1976}} p. 54, § 12.11.3

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

• {{Cite book|last=Axler|first=Sheldon|title=Linear Algebra Done Right|publisher= Springer|year=2015|isbn=978-3-319-11079-0|edition=3rd|series=Undergraduate Texts in Mathematics|location=|pages=|author-link=Sheldon Axler}}
• {{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis|publisher=Academic Press|year=1976|volume=2|pages=|isbn=978-0122155024}}
• {{Cite book|last=Halmos|first=Paul Richard|title=Finite-Dimensional Vector Spaces|publisher= Springer|year=1974|isbn=0-387-90093-4|edition=2nd|series=Undergraduate Texts in Mathematics|volume=|location=|pages=|author-link=Paul Halmos|orig-year=1958}}
• {{Cite book|last=Katznelson|first=Yitzhak|title=A (Terse) Introduction to Linear Algebra|last2=Katznelson|first2=Yonatan R.|publisher=American Mathematical Society|isbn=978-0-8218-4419-9|volume=|publication-date=2008|pages=|author-link=Yitzhak Katznelson}}
• {{Cite book|last=Roman|first=Steven|title=Advanced Linear Algebra|publisher=Springer|year=2005|isbn=0-387-24766-1|edition=2nd|series=Graduate Texts in Mathematics|location=|pages=|author-link=Steven Roman}}

{{Linear algebra}}

Category:Functional analysis

Category:Linear algebra

Space (linear algebra)