Quotient space (linear algebra) explained

V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read "V mod N " or "V by N ").

Definition

Formally, the construction is as follows.^[1] Let

V be a vector space over a field K , and let N be a subspace of V . We define an equivalence relation \sim on V by stating that x \sim y if . That is, x is related to y if one can be obtained from the other by adding an element of N . From this definition, one can deduce 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 often denoted[x] = x + N since it is given by[x] = \{ x + n: n \in N \}

The quotient space

V/N is then defined as V/ _{\sim} , the set of all equivalence classes induced by \sim on V . Scalar multiplication and addition are defined on the equivalence classes by^{[2] [3]} \alpha [x] = [\alpha x] for all \alpha \in 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 K with N being the zero class, [ 0 ] .

The mapping that associates to 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 ^[4]

Examples

Lines in Cartesian Plane

Let 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 . The subspace, identified with R^m, consists of all n-tuples such that the last n − m entries are zero: . 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

l{P} _{3(R)} be the vector space of all cubic polynomials over the real numbers. Then l{P} _{3(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 \{x ^{3} + a x ^{2} - 2 x + 3 : a \in R \} , while another element of the quotient space is \{a x ^{2} + 2 . 7 x : a \in R \} .

General Subspaces

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

V = U ⊕ 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

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]}

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

\| [x] \| _{X/M} = inf _{m} \|x - m\| _{X} = inf _{m} \|x + m\| _{X} = inf _{y\in} \|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 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 where A is an index set. Let M be a closed subspace, and define seminorms q_α on X/M by

q _{\alpha([x])} = inf _{v\in} 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. pp. 33-34 §§ 21-22
2. p. 9 § 1.2.4
3. p. 75-76, ch. 3
4. p. 95, § 3.83
5. p. 34, § 22, Theorem 1
6. p. 97, § 3.89
7. p. 34, § 22, Theorem 2
8. p. 65, § 12.14.8
9. p. 54, § 12.11.3

Sources

• Book: Axler, Sheldon. Linear Algebra Done Right. Springer. 2015. 978-3-319-11079-0. 3rd. Undergraduate Texts in Mathematics. Sheldon Axler.
  • Book: Halmos, Paul Richard. Finite-Dimensional Vector Spaces. Springer. 1974. 0-387-90093-4. 2nd. Undergraduate Texts in Mathematics. Paul Halmos. 1958.
• Book: Katznelson, Yitzhak. A (Terse) Introduction to Linear Algebra. Katznelson. Yonatan R.. American Mathematical Society. 978-0-8218-4419-9. 2008. Yitzhak Katznelson.
• Book: Roman, Steven. Advanced Linear Algebra. Springer. 2005. 0-387-24766-1. 2nd. Graduate Texts in Mathematics. Steven Roman.