Quotient Spaces — Mathematics Knowledge System

Overview

A quotient space V/W is formed from a vector space V by collapsing a subspace W to zero — treating all vectors that differ by an element of W as equivalent. Quotient spaces are fundamental in abstract algebra and appear in the first isomorphism theorem: every linear map T: V → U factors through the quotient V/ker(T).

Intuition

Imagine V = R^2 and W = the x-axis. The quotient V/W is obtained by identifying all points that have the same y-coordinate — we collapse each vertical line to a single point. The resulting space is effectively a copy of R (parameterized by y). More generally, V/W asks: 'what's left of V once we consider W as zero?' Each coset v + W is a translate of W, and the quotient space is the set of all such translates.

Formal Definition

Definition

Let V be a vector space over F and W a subspace. Two vectors u, v ∈ V are W-equivalent if u - v ∈ W. The quotient space is:

V/W = \{ v + W \mid v \in V \}

Elements of V/W are cosets v+W = {v+w : w ∈ W}.

Quotient space

(u+W) + (v+W) = (u+v)+W, \qquad \alpha(v+W) = (\alpha v)+W

Vector space operations

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

Dimension formula

Notation

Notation	Meaning
$V/W$	Quotient space of V modulo the subspace W
$v + W$	Coset of v in V/W: the equivalence class of v
$\pi$	Canonical projection π: V → V/W, v ↦ v+W

Theorems

Theorem 1: First Isomorphism Theorem

\text{If } T: V \to U \text{ is linear, then } V/\ker(T) \cong \operatorname{im}(T).

Worked Examples

1
Two vectors (x_1,y_1,z_1) and (x_2,y_2,z_2) are W-equivalent iff their difference lies in W, i.e., z_1 = z_2.
$(x_1,y_1,z_1)-(x_2,y_2,z_2) \in W \iff z_1 - z_2 = 0 \iff z_1 = z_2$
2
So each coset is determined by z alone: the coset (x,y,z)+W = {(a,b,z): a,b ∈ R}.
3
V/W is parameterized by z ∈ R, so V/W ≅ R.
$\dim(V/W) = \dim(V) - \dim(W) = 3 - 2 = 1$

✓ Answer

V/W ≅ R, dimension 1. Each coset is a horizontal plane at height z.

Practice Problems

Mediumfree response

Let V = R^4 and W = span{(1,0,0,0),(0,1,0,0)}. What is dim(V/W)?

Mediumfree response

Let T: R^3 → R^3 be T(x,y,z) = (x-y, y-z, 0). Find ker(T), describe V/ker(T), and identify im(T) using the First Isomorphism Theorem.

Common Mistakes

Common Mistake

Thinking v+W is a single vector rather than a set.

A coset v+W is a whole affine subspace — the set {v+w : w ∈ W}. It is an element of V/W but as a set of vectors in V.

Quiz

Summary

V/W collapses the subspace W to zero; elements are cosets v+W.
dim(V/W) = dim(V) - dim(W).
The canonical projection π: V → V/W, v ↦ v+W, is linear and surjective with kernel W.
The First Isomorphism Theorem: V/ker(T) ≅ im(T) for any linear T: V → U.
Quotient spaces are used to factor out redundancy and simplify arguments about linear maps.

References

WebsiteWikipedia — Quotient space (linear algebra)