# Quotient space (linear algebra)

__: Vector space consisting of affine subsets__

**Short description**

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.^{[1]} 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^{[2]}^{[3]}

- [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].^{[4]}

## Examples

### Lines in Cartesian Plane

Let *X* = **R**^{2} 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 **R**^{3} 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 **R**^{n} by the subspace spanned by the first *m* standard basis vectors. The space **R**^{n} consists of all *n*-tuples of real numbers (*x*_{1}, ..., *x*_{n}). The subspace, identified with **R**^{m}, consists of all *n*-tuples such that the last *n* − *m* entries are zero: (*x*_{1}, ..., *x*_{m}, 0, 0, ..., 0). Two vectors of **R**^{n} 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 **R**^{n}/**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*.^{[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

- [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*:^{[6]}^{[7]}

- [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.^{[citation needed]}

### 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

- [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*.^{[9]}

## See also

## References

- ↑ (Halmos 1974) pp. 33-34 §§ 21-22
- ↑ (Katznelson Katznelson) p. 9 § 1.2.4
- ↑ (Roman 2005) p. 75-76, ch. 3
- ↑ (Axler 2015) p. 95, § 3.83
- ↑ (Halmos 1974) p. 34, § 22, Theorem 1
- ↑ (Axler 2015) p. 97, § 3.89
- ↑ (Halmos 1974) p. 34, § 22, Theorem 2
- ↑ (Dieudonné 1976) p. 65, § 12.14.8
- ↑ (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*,**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.

Original source: https://en.wikipedia.org/wiki/Quotient space (linear algebra).
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