Quotient space (linear algebra)

In linear algebra, the quotient of a vector space $V$ by a subspace $U$ is a vector space obtained by "collapsing" $U$ to zero. The space obtained is called a quotient space and is denoted $V/U$ (read "$V$ mod $U$" or "$V$ by $U$"). Alternatively phrased, the quotient space $V/U$ is the set of all affine subsets of $V$ which are parallel to $U$. A quotient space $A/U$ can also be defined for an affine space $A$ by a linear space $U$.

Definition

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

The equivalence class – or, in this case, the coset – of $v$ is defined as

$[v]:=\{w:v-w\in U\}$.

Equivalently, $[v]=\{v+u:u\in U\}$, so it is often denoted using the shorthand $v+U$.

The quotient space $V/U$ is then defined as $V/\sim$, the set of all equivalence classes induced by $\sim$ on $U$. Scalar multiplication and addition are defined on the equivalence classes by

• $\alpha [x]=[\alpha x]$ 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 $V/U$ into a vector space over $𝕂$ with $U$ being the zero class, $[0]$.

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

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 (relative positions) 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 ${𝒫}_3(ℝ)$ be the vector space of all cubic polynomials over the real numbers. Then ${𝒫}_3(ℝ)/⟨x^2⟩$ 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 $\{x^3+a{x}^2-2x+3:a\in ℝ\}$, while another element of the quotient space is $\{a{x}^2+2.7x:a\in ℝ\}$.

General Subspaces

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

$V=U\oplus W$

then the quotient space V/U is naturally isomorphic to W.

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:

$\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}(U)=\dim (V/U)=\dim (V)-\dim (U).$

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).

Isomorphism Theorems

First Isomorphism Theorem

Let $V,W$ be $𝕂$-Vector Spaces and $T:V\to W$ linear. Define the map $\bar{T}:V/\ker T\to \mathrm{im}(T)$ by $\bar{T}([v])=T(v).$ Then $\bar{T}$ is well-defined and an isomorphism.

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

$\Vert [x]{\Vert }_{X/M}=\underset{m\in M}{inf}\Vert x-m{\Vert }_X=\underset{m\in M}{inf}\Vert x+m{\Vert }_X=\underset{y\in [x]}{inf}\Vert y{\Vert }_X.$

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 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. 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

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

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.

See also

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

Sources

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