# Chapter 44. Quotient Spaces and Quotient Operators

# 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. ([en.wikipedia.org](https://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29?utm_source=chatgpt.com))

## 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:

If

$$
v-w\in U,
$$

then

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

Transitivity:

If

$$
v-w\in U
$$

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,

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

So

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

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:

If

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

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. ([math.libretexts.org](https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_%28Schilling_Nachtergaele_and_Lankham%29/07%3A_Quotient_Spaces/7.01%3A_Quotient_Spaces?utm_source=chatgpt.com))

## 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

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

by

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

This is well-defined because if

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

then

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

so

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

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

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

by

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

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:

1. Choose a substructure.
2. Declare elements differing by that substructure to be equivalent.
3. Form equivalence classes.
4. 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.

If

$$
T:V\to W
$$

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.