Chapter 44. Quotient Spaces and Quotient Operators

Quotient spaces allow vectors that differ by elements of a subspace to be treated as equivalent. They formalize the idea of collapsing a subspace to zero.

If (V) is a vector space and (U\subseteq V) is a subspace, the quotient space

$$ V/U $$

consists of equivalence classes of vectors modulo (U). Quotient operators arise when a linear operator preserves the subspace (U), allowing the operator to act naturally on the quotient space. Quotient constructions are fundamental in linear algebra, abstract algebra, topology, and geometry.

44.1 Motivation

Suppose we want to treat vectors that differ by an element of a subspace (U) as essentially the same.

For example, in (\mathbb{R}^3), let

$$ U= \left{ \begin{bmatrix} 0\ 0\ z \end{bmatrix} :z\in\mathbb{R} \right}. $$

This is the (z)-axis.

Two vectors

$$ \begin{bmatrix} x_1\ y_1\ z_1 \end{bmatrix}, \qquad \begin{bmatrix} x_2\ y_2\ z_2 \end{bmatrix} $$

differ by an element of (U) exactly when

$$ x_1=x_2, \qquad y_1=y_2. $$

Thus vectors with the same (x)- and (y)-coordinates become identified.

The quotient space

$$ \mathbb{R}^3/U $$

therefore behaves like the plane (\mathbb{R}^2). The entire (z)-direction has been collapsed.

44.2 Equivalence Relation

Let (V) be a vector space and let (U\subseteq V) be a subspace.

Define a relation on (V) by

$$ v\sim w $$

if and only if

$$ v-w\in U. $$

This is an equivalence relation.

Reflexivity:

$$ v-v=0\in U. $$

Symmetry:

$$ v-w\in U, $$

then

$$ w-v=-(v-w)\in U. $$

Transitivity:

$$ v-w\in U $$

and

$$ w-z\in U, $$

then

$$ v-z=(v-w)+(w-z)\in U. $$

Thus vectors are equivalent exactly when their difference lies in (U).

44.3 Cosets

The equivalence class of (v\in V) is

$$ v+U={v+u:u\in U}. $$

This set is called the coset of (U) determined by (v).

The quotient space

$$ V/U $$

is the set of all cosets:

$$ V/U={v+U:v\in V}. $$

Each coset is an affine copy of the subspace (U).

Two vectors determine the same coset exactly when they differ by an element of (U):

$$ v+U=w+U \iff v-w\in U. $$

Thus the quotient space partitions (V) into parallel copies of (U).

44.4 Vector Space Structure

The quotient space (V/U) becomes a vector space by defining addition and scalar multiplication on cosets.

Addition:

$$ (v+U)+(w+U)=(v+w)+U. $$

Scalar multiplication:

$$ c(v+U)=(cv)+U. $$

These definitions must be well-defined. That is, they must not depend on the chosen representatives.

Suppose

$$ v+U=v'+U $$

and

$$ w+U=w'+U. $$

Then

$$ v-v'\in U, \qquad w-w'\in U. $$

Therefore

$$ (v+w)-(v'+w')=(v-v')+(w-w')\in U. $$

Hence

$$ (v+w)+U=(v'+w')+U. $$

Similarly,

$$ cv-cv'=c(v-v')\in U. $$

$$ cv+U=cv'+U. $$

Thus the operations are well-defined.

44.5 Zero Element

The zero vector of (V/U) is

$$ U=0+U. $$

Indeed,

$$ (v+U)+U=v+U. $$

A coset equals zero exactly when its representative lies in (U):

$$ v+U=U \iff v\in U. $$

Thus the subspace (U) itself becomes the zero vector in the quotient space.

This is why quotient spaces are often described as collapsing (U) to zero.

44.6 Example in (\mathbb{R}^2)

Let

$$ U= \operatorname{span} \left{ \begin{bmatrix} 1\ 0 \end{bmatrix} \right}. $$

This is the (x)-axis.

Two vectors

$$ \begin{bmatrix} x_1\ y_1 \end{bmatrix}, \qquad \begin{bmatrix} x_2\ y_2 \end{bmatrix} $$

belong to the same coset exactly when

$$ y_1=y_2. $$

Indeed,

$$ \begin{bmatrix} x_1\ y_1 \end{bmatrix} - \begin{bmatrix} x_2\ y_2 \end{bmatrix} = \begin{bmatrix} x_1-x_2\ y_1-y_2 \end{bmatrix} $$

lies in (U) exactly when

$$ y_1-y_2=0. $$

Thus each coset corresponds to a horizontal line.

The quotient space

$$ \mathbb{R}^2/U $$

therefore behaves like the (y)-axis. The horizontal direction has been collapsed.

44.7 Dimension Formula

If (V) is finite-dimensional and (U\subseteq V), then

$$ \dim(V/U)=\dim(V)-\dim(U). $$

To prove this, choose a basis

$$ (u_1,\ldots,u_k) $$

for (U), and extend it to a basis of (V):

$$ (u_1,\ldots,u_k,v_1,\ldots,v_m). $$

Then

$$ \dim(V)=k+m. $$

The cosets

$$ v_1+U,\ldots,v_m+U $$

form a basis of (V/U).

Spanning:

Every vector in (V) has the form

$$ u+a_1v_1+\cdots+a_mv_m. $$

Its coset equals

$$ a_1(v_1+U)+\cdots+a_m(v_m+U). $$

Linear independence:

$$ a_1(v_1+U)+\cdots+a_m(v_m+U)=U, $$

then

$$ a_1v_1+\cdots+a_mv_m\in U. $$

Since the full list is a basis of (V), this forces

$$ a_1=\cdots=a_m=0. $$

Thus

$$ \dim(V/U)=m=\dim(V)-\dim(U). $$

This is the dimension formula for quotient spaces.

44.8 Canonical Projection

The canonical projection map is

$$ \pi:V\to V/U, \qquad \pi(v)=v+U. $$

This map sends each vector to its coset.

The map (\pi) is linear:

$$ \pi(v+w)=(v+w)+U=(v+U)+(w+U), $$

and

$$ \pi(cv)=cv+U=c(v+U). $$

The kernel of (\pi) is

$$ \ker(\pi)=U. $$

Indeed,

$$ \pi(v)=U $$

exactly when

$$ v\in U. $$

The image of (\pi) is all of (V/U).

Thus (\pi) is a surjective linear map with kernel (U).

44.9 First Isomorphism Theorem

Let

$$ T:V\to W $$

be linear.

Then

$$ V/\ker(T)\cong \operatorname{im}(T). $$

This is the first isomorphism theorem for vector spaces.

Define

$$ \Phi:V/\ker(T)\to \operatorname{im}(T) $$

$$ \Phi(v+\ker(T))=T(v). $$

This is well-defined because if

$$ v+\ker(T)=w+\ker(T), $$

then

$$ v-w\in \ker(T), $$

$$ T(v-w)=0, $$

hence

$$ T(v)=T(w). $$

The map is linear, surjective, and injective. Therefore it is an isomorphism.

This theorem shows that quotient spaces naturally arise from linear maps.

44.10 Quotient Operators

Let

$$ T:V\to V $$

be linear, and let (U\subseteq V) be invariant under (T).

Then (T) induces a linear operator on the quotient space:

$$ \widetilde{T}:V/U\to V/U, $$

defined by

$$ \widetilde{T}(v+U)=T(v)+U. $$

This is called the quotient operator induced by (T).

The invariance of (U) is essential. Without it, the definition may not be well-defined.

44.11 Why Invariance Is Needed

Suppose

$$ v+U=w+U. $$

Then

$$ v-w\in U. $$

To show the quotient operator is well-defined, we need

$$ T(v)+U=T(w)+U. $$

This means

$$ T(v)-T(w)\in U. $$

But

$$ T(v)-T(w)=T(v-w). $$

So we need

$$ T(v-w)\in U $$

whenever

$$ v-w\in U. $$

This is exactly the condition

$$ T(U)\subseteq U. $$

Thus quotient operators exist precisely when the subspace is invariant.

44.12 Example of a Quotient Operator

Let

$$ T:\mathbb{R}^2\to\mathbb{R}^2 $$

be defined by

$$ T \begin{bmatrix} x\ y \end{bmatrix} = \begin{bmatrix} x+y\ y \end{bmatrix}. $$

Its matrix is

$$ A= \begin{bmatrix} 1&1\ 0&1 \end{bmatrix}. $$

Let

$$ U= \operatorname{span} \left{ \begin{bmatrix} 1\ 0 \end{bmatrix} \right}. $$

Then (U) is invariant because

$$ T \begin{bmatrix} x\ 0 \end{bmatrix} = \begin{bmatrix} x\ 0 \end{bmatrix}. $$

The quotient space (V/U) behaves like the (y)-axis.

Now

$$ T \begin{bmatrix} x\ y \end{bmatrix} = \begin{bmatrix} x+y\ y \end{bmatrix}. $$

Modulo (U), the first coordinate disappears. Thus the quotient operator acts by

$$ y\mapsto y. $$

So the induced operator on (V/U) is the identity.

44.13 Matrix Form of a Quotient Operator

Suppose (U\subseteq V) is invariant under (T).

Choose a basis

$$ (u_1,\ldots,u_k,v_1,\ldots,v_m) $$

such that

$$ (u_1,\ldots,u_k) $$

is a basis of (U).

Then the matrix of (T) has block upper triangular form

$$ \begin{bmatrix} A&B\ 0&C \end{bmatrix}. $$

The block (A) represents the restriction of (T) to (U).

The block (C) represents the quotient operator on (V/U).

Thus quotient operators naturally appear inside block triangular decompositions.

44.14 Quotient by the Kernel

Let

$$ T:V\to W $$

be linear.

The quotient space

$$ V/\ker(T) $$

removes exactly the directions invisible to (T).

Indeed, vectors (v) and (w) lie in the same coset exactly when

$$ v-w\in \ker(T). $$

This means

$$ T(v)=T(w). $$

Thus the quotient identifies vectors that have the same image under (T).

The first isomorphism theorem says that after collapsing the kernel, the map becomes injective.

44.15 Quotient Spaces and Geometry

Quotient spaces often reduce dimension by ignoring selected directions.

Examples:

Original space	Subspace collapsed	Quotient behaves like
(\mathbb{R}^3)	(z)-axis	(\mathbb{R}^2)
(\mathbb{R}^2)	(x)-axis	(\mathbb{R})
Polynomial space	Multiples of (x^2)	Lower-degree polynomials

The quotient construction keeps only information transverse to the chosen subspace.

44.16 Quotient and Direct Sum

Suppose

$$ V=U\oplus W. $$

Then every vector can be written uniquely as

$$ u+w. $$

In this case,

$$ V/U\cong W. $$

Indeed, every coset has a unique representative in (W).

Define

$$ \Phi:W\to V/U $$

$$ \Phi(w)=w+U. $$

This map is linear, injective, and surjective.

Thus quotient spaces are closely related to complementary subspaces.

However, quotient spaces do not require a chosen complement. The quotient construction works even when no natural complement exists.

44.17 Quotient Spaces and Duality

If (U\subseteq V), then functionals vanishing on (U) correspond naturally to linear functionals on (V/U).

Indeed, if

$$ f:V\to F $$

satisfies

$$ f(u)=0 $$

for every (u\in U), then (f) depends only on the coset (v+U).

Thus there exists a unique functional

$$ \widetilde{f}:V/U\to F $$

such that

$$ \widetilde{f}(v+U)=f(v). $$

This connection is important in dual spaces, annihilators, and functional analysis.

44.18 Quotient and Polynomial Operators

Suppose (U) is invariant under (T). Then every polynomial in (T) also induces a quotient operator.

Indeed,

$$ p(T)(v+U)=p(T)(v)+U. $$

This is well-defined because invariance under (T) implies invariance under every power of (T), and therefore under every polynomial in (T).

Thus quotient constructions are compatible with operator algebra.

44.19 Quotient Spaces in Module Theory

Quotient spaces are vector-space versions of quotient modules and quotient groups.

The construction always has the same form:

Choose a substructure.
Declare elements differing by that substructure to be equivalent.
Form equivalence classes.
Define induced operations.

In vector spaces, the substructure is a subspace. In groups, it is a normal subgroup. In rings, it is an ideal.

The linear-algebra quotient construction is therefore part of a broader algebraic pattern.

44.20 Quotient Spaces and Coordinates

Suppose

$$ V=U\oplus W. $$

Then every vector has coordinates

$$ (u,w). $$

Passing to the quotient (V/U) removes the (U)-coordinates and keeps only the (W)-coordinates.

This viewpoint explains why quotient spaces often behave like complementary subspaces, even though no complement is built into the definition.

The quotient remembers only directions not absorbed into (U).

44.21 Universal Property

The quotient map

$$ \pi:V\to V/U $$

has the following universal property.

$$ T:V\to W $$

is linear and

$$ U\subseteq \ker(T), $$

then there exists a unique linear map

$$ \widetilde{T}:V/U\to W $$

such that

$$ T=\widetilde{T}\circ \pi. $$

Diagrammatically,

$$ V \overset{\pi}{\longrightarrow} V/U \overset{\widetilde{T}}{\longrightarrow} W. $$

This property characterizes the quotient space abstractly and explains why quotient constructions appear naturally throughout algebra.

44.22 Summary

Let (U\subseteq V) be a subspace.

The quotient space

$$ V/U $$

consists of cosets

$$ v+U. $$

Two vectors are identified when their difference lies in (U).

The quotient space has vector operations

$$ (v+U)+(w+U)=(v+w)+U $$

and

$$ c(v+U)=cv+U. $$

Its dimension satisfies

$$ \dim(V/U)=\dim(V)-\dim(U). $$

The canonical projection

$$ \pi:V\to V/U $$

has kernel (U).

If (T:V\to V) preserves (U), then (T) induces a quotient operator

$$ \widetilde{T}:V/U\to V/U. $$

Quotient spaces formalize the idea of collapsing a subspace to zero. They are fundamental in linear algebra, operator theory, geometry, algebra, and topology.