Quotient vector space · aiwiki.page

A quotient vector space is a vector space formed by identifying vectors that differ by an element of a specified linear subspace. If $V$ is a vector space over a field $F$ and $W\subseteq V$ is a subspace, the quotient is written $V/W$ . Its elements are sets of vectors rather than individual vectors of $V$ . Addition and scalar multiplication are inherited from $V$ , making the construction a way to disregard differences lying in $W$ . (heil.math.gatech.edu)

Definition and vector-space operations

Define an equivalence relation on $V$ by

v\sim u\quad\Longleftrightarrow\quad v-u\in W.

The equivalence class of $v$ is its coset

[v]=v+W=\{v+w:w\in W\}.

Consequently,

V/W=\{v+W:v\in V\}.

Two cosets are equal precisely when their representatives differ by an element of $W$ ; otherwise they are disjoint. Thus the cosets partition $V$ . (heil.math.gatech.edu)

The quotient operations are

(v+W)+(u+W)=(v+u)+W,\qquad a(v+W)=av+W.

They are independent of the representatives chosen. For example, replacing $v$ and $u$ by $v+w_1$ and $u+w_2$ changes their sum by $w_1+w_2$ , which remains in $W$ . Closure under scalar multiplication provides the corresponding verification for scalar multiples. The vector-space axioms then follow from those of $V$ . The zero element is the entire coset $W$ , and the additive inverse of $v+W$ is $-v+W$ . (math.uh.edu)

The notation $V/W$ does not denote numerical division or the removal of $W$ as a set. Every vector of $W$ becomes the same zero element, while vectors outside $W$ are grouped according to their differences. (heil.math.gatech.edu)

Canonical projection and universal property

The canonical projection

q:V\longrightarrow V/W,\qquad q(v)=v+W

is a linear map and a surjective function. Its kernel is exactly $W$ . (heil.math.gatech.edu)

Its defining mapping property is the universal property of the quotient: whenever a linear map $T:V\to U$ satisfies $W\subseteq\ker T$ , there is a unique linear map

\overline T:V/W\longrightarrow U

such that $T=\overline T\circ q$ . Explicitly,

\overline T(v+W)=T(v).

This is well-defined because $T(v+w)=T(v)$ for every $w\in W$ ; uniqueness follows because every quotient element has a representative. Equivalently, a linear map descends to the quotient precisely when it vanishes on $W$ . (math.uh.edu)

Taking $W=\ker T$ gives the first isomorphism theorem:

V/\ker T\cong\operatorname{im}T, \qquad v+\ker T\longmapsto T(v).

The induced map is an isomorphism onto the image of $T$ . The quotient therefore retains exactly the distinctions detected by $T$ . (math.mit.edu)

Dimension, bases, and complements

For finite-dimensional $V$ ,

\dim(V/W)=\dim V-\dim W.

This dimension is called the codimension of $W$ in $V$ . The extreme cases are $V/\{0\}\cong V$ and $V/V=\{0\}$ . (math.mit.edu)

A concrete basis is obtained by extending a basis $w_1,\ldots,w_k$ of $W$ to a basis

w_1,\ldots,w_k,v_{k+1},\ldots,v_n

of $V$ . Then

v_{k+1}+W,\ldots,v_n+W

form a basis of $V/W$ . Their linear independence follows because a linear combination lying in $W$ would give a dependence among the extended basis vectors. Their cosets span the quotient because the $W$ -components contribute zero. (math.mit.edu)

The added vectors span a complement $C$ , giving a direct sum $V=W\oplus C$ . Restricting $q$ to $C$ yields an isomorphism $C\to V/W$ . However, this complement generally depends on choices; the quotient itself requires no choice of representatives. Applying the dimension formula to $\ker T$ yields the rank–nullity theorem. (math.mit.edu)

Geometric interpretation and example

Over the real numbers, each coset is an affine subspace parallel to $W$ . If $W$ is a line in a plane, the quotient consists of all lines parallel to that line, with each whole line treated as one vector. (heil.math.gatech.edu)

For example, take

V=\mathbb R^2,\qquad W=\{(x,0):x\in\mathbb R\}.

The coset of $(a,b)$ is the horizontal line

(a,b)+W=\{(x,b):x\in\mathbb R\}.

Its class depends only on $b$ , so

\mathbb R^2/W\longrightarrow\mathbb R,\qquad (a,b)+W\longmapsto b

is a linear isomorphism. Adding quotient vectors corresponds to adding line heights, and multiplying by a scalar multiplies the height. (math.mit.edu)

Normed and Hilbert-space quotients

If $V$ is a normed vector space, the quotient carries the seminorm

\|v+W\|_{V/W}=\inf_{w\in W}\|v-w\|_V.

It measures the distance from a representative to $W$ . This is a genuine norm exactly when $W$ is closed. If $V$ is a Banach space and $W$ is closed, the quotient is also complete and hence a Banach space. (people.math.ethz.ch)

For a closed subspace of a Hilbert space, every coset has a unique representative in the orthogonal complement $W^\perp$ . The identification $V/W\cong W^\perp$ is isometric: the representative is obtained by orthogonal projection onto $W^\perp$ . Unlike an arbitrary algebraic complement, this choice is determined by the given inner-product structure. (people.math.ethz.ch)