Quotient vector space

1 Basic definition and cosets

1.1 Subspace and equivalence relation

1.1.1 Cosets as equivalence classes

Let \(V\) be a vector space and \(W \le V\) a subspace. For each \(v \in V\), the set \[ v+W=\{v+w : w\in W\} \] is called the coset of \(W\) determined by \(v\). The quotient vector space \(V/W\) consists of all such cosets.

The key idea is that cosets bundle together vectors that differ by something in \(W\), so they represent the “same” element of the quotient.

1.1.2 Well-defined equivalence via \(v-u \in W\)

Define a relation \(\sim\) on \(V\) by \[ u \sim v \quad \Longleftrightarrow \quad v-u \in W. \] This relation is an equivalence relation: it is reflexive because \(v-v=0\in W\), symmetric because if \(v-u\in W\) then \(u-v=-(v-u)\in W\), and transitive because \((v-u)+(u-t)=v-t\in W\). The equivalence class of \(v\) is precisely \(v+W\).

Accordingly, elements of \(V/W\) are the equivalence classes of \(\sim\), written as cosets \(v+W\).

1.2 Construction of \(V/W\)

1.2.1 Defining addition on cosets

Addition in the quotient is defined by \[ (v+W) + (u+W) := (v+u) + W. \] Because each coset represents an equivalence class, this definition must not depend on which representatives are chosen. The result is the same regardless of the particular representatives of the cosets.

1.2.2 Defining scalar multiplication on cosets

Scalar multiplication is defined by \[ \alpha\,(v+W) := (\alpha v) + W \quad (\alpha \in \mathbb{F}), \] where \(\mathbb{F}\) is the underlying field of \(V\). As with addition, it is required that this rule produces a well-defined coset independent of the chosen representative.

1.2.3 Proof of vector space axioms

With addition and scalar multiplication defined, one checks the vector space axioms by translating them back to corresponding axioms in \(V\). For instance:

• Commutativity of addition follows since \((v+u)+W=(u+v)+W\).
• Associativity follows from \(((v+u)+t)+W=(v+(u+t))+W\).
• Distributive laws and compatibility with scalar multiplication follow similarly from properties in \(V\).

The existence of an additive identity and inverses comes from cosets: the identity is \(0+W=W\), and the inverse of \(v+W\) is \((-v)+W\).

2 Structural properties

2.1 Canonical projection map

2.1.1 The map \(\pi: V \to V/W\)

There is a natural surjection \[ \pi: V \to V/W,\qquad \pi(v)=v+W, \] sending each vector to its coset. This map is often called the canonical projection because it “collapses” all vectors in the same coset into one quotient element.

2.1.2 Kernel of the projection

The kernel of \(\pi\) is \[ \ker(\pi)=\{v\in V : \pi(v)=W\}. \] But \(\pi(v)=W\) precisely when \(v+W=W\), which happens exactly when \(v\in W\). Hence \[ \ker(\pi)=W. \]

2.1.3 Image and surjectivity

Every coset has the form \(v+W=\pi(v)\), so \(\pi\) is surjective: \[ \operatorname{im}(\pi)=V/W. \]

2.2 Quotient map universality

2.2.1 Factorization of linear maps

A quotient space is characterized by a universal property: linear maps that vanish on \(W\) factor uniquely through the projection \(\pi\).

Concretely, suppose \(f:V\to U\) is linear and satisfies \(W\subseteq \ker f\). Then for each coset \(v+W\), one may define \[ \tilde f(v+W)=f(v). \] This is well-defined because if \(v+W=u+W\), then \(v-u\in W\subseteq\ker f\), so \(f(v)=f(u)\). With that definition, \(\tilde f:V/W\to U\) is linear and satisfies \(f=\tilde f\circ \pi\).

2.2.2 Characterizing when a map factors through \(V/W\)

More generally, a linear map \(f:V\to U\) factors through \(V/W\) (meaning \(f=\tilde f\circ \pi\) for some \(\tilde f\)) exactly when \(f\) is constant on cosets; equivalently, exactly when \(W\subseteq \ker f\). This condition captures precisely the idea that vectors differing by an element of \(W\) are indistinguishable under \(f\).

3 Relation to linear transformations

3.1 Quotients via kernels

3.1.1 From \(T: V \to U\) to \(V/\ker T\)

Given a linear transformation \(T:V\to U\), its kernel \(\ker T\) is a subspace of \(V\). The quotient \(V/\ker T\) measures the part of \(V\) that remains visible after collapsing vectors that \(T\) sends to zero.

There is an induced structure: elements of \(V/\ker T\) correspond to cosets where all vectors differing by something in \(\ker T\) are identified.

3.1.2 Induced linear map on the quotient

Define a map \(\tilde T:V/\ker T \to \operatorname{im}T\) by \[ \tilde T(v+\ker T)=T(v). \] This is well-defined because if \(v+\ker T=u+\ker T\), then \(v-u\in\ker T\), so \(T(v)=T(u)\). The map is linear and surjective onto \(\operatorname{im}T\). It is also injective, since \(\tilde T(v+\ker T)=0\) implies \(T(v)=0\), hence \(v\in\ker T\).

Thus \(V/\ker T\) is naturally isomorphic to \(\operatorname{im}T\).

3.2 First Isomorphism Theorem (vector-space form)

3.2.1 Statement and interpretation

The vector-space form of the First Isomorphism Theorem states: for a linear map \(T:V\to U\), \[ V/\ker T \cong \operatorname{im}T. \] Interpretation: the quotient identifies vectors that are indistinguishable to \(T\), and the image records exactly what survives.

3.2.2 Consequences for dimensions

When \(V\) and \(W\) are finite-dimensional, the theorem yields dimension relationships. In particular, \[ \dim(V/\ker T)=\dim(\operatorname{im}T). \] Combined with rank–nullity, this gives \[ \dim V=\dim(\ker T)+\dim(\operatorname{im}T). \] Since \(\dim(\ker T)\) measures the lost information under \(T\), the quotient restores the remaining degrees of freedom.

4 Dimension and basis techniques

4.1 Dimension formula

4.1.1 \(\dim(V/W)=\dim V-\dim W\) (finite-dimensional case)

If \(V\) is finite-dimensional and \(W\le V\) is a subspace, then \[ \dim(V/W)=\dim V-\dim W. \] A standard approach uses a basis \(w_1,\dots,w_k\) for \(W\) and extends it to a basis of \(V\). The quotient then has basis elements corresponding to the additional directions beyond \(W\).

4.2 Bases in quotient spaces

4.2.1 Extending a basis of \(W\) to a basis of \(V\)

Let \(\{w_1,\dots,w_k\}\) be a basis for \(W\). If \(\dim V=n\), one can extend this to a basis of \(V\), \[ \{w_1,\dots,w_k, v_{k+1},\dots,v_n\}. \] The extra vectors \(v_{k+1},\dots,v_n\) represent directions in \(V\) not contained in \(W\).

4.2.2 Constructing a basis of \(V/W\)

Under the extension above, the cosets \[ \{v_{k+1}+W,\dots,v_n+W\} \] form a basis of \(V/W\). Intuitively, every vector in \(V\) can be written as a combination of the basis vectors; modding out by \(W\) removes the components along \(w_1,\dots,w_k\), leaving only coefficients for the \(v_{k+1},\dots,v_n\) directions.

4.3 Representing elements and coordinates

4.3.1 Choosing representatives

An element of \(V/W\) is a coset \(v+W\), but calculations typically require picking a representative \(v\). Different choices of representatives correspond to adding an element of \(W\). In practice, one chooses a convenient representative, often one that uses only the “complement” directions from a basis construction.

4.3.2 Working with quotient coordinates

Once a basis for \(V/W\) is fixed (as in the previous section), any coset can be expressed uniquely as a linear combination of the basis cosets. Concretely, if \(\{v_{k+1}+W,\dots,v_n+W\}\) is a basis, then for each \(v\in V\) one determines coefficients \(\alpha_{k+1},\dots,\alpha_n\) such that \[ v+W=\sum_{j=k+1}^n \alpha_j (v_j+W). \] These coefficients serve as quotient coordinates; changing the representative \(v\) by an element of \(W\) does not alter the resulting coefficients.

5 Subspaces and lattice behavior

5.1 Correspondence with subspaces of \(V\)

5.1.1 Subspaces containing \(W\)

There is a natural correspondence between subspaces of \(V/W\) and subspaces of \(V\) that contain \(W\). If \(U\) is a subspace of \(V\) with \(W\subseteq U\), one can form the quotient \(U/W\), which is a subspace of \(V/W\).

Conversely, given a subspace \(S\le V/W\), its preimage under the projection \(\pi\) is a subspace of \(V\) containing \(W\). This creates an order-preserving relationship between the two lattices of subspaces.

5.1.2 Induced subspaces in \(V/W\)

For \(U\) with \(W\subseteq U\), the set \[ U/W=\{u+W : u\in U\} \] is a subspace of \(V/W\). Operations such as sums and intersections within \(U/W\) mirror corresponding operations among \(U\)-subspaces in \(V\).

5.2 Quotient of a quotient

5.2.1 The double-quotient idea

Quotients can be taken in stages. After collapsing by \(W_1\), one may further collapse by the image of another subspace \(W_2\) (with \(W_1\subseteq W_2\)).

This staged approach formalizes the idea that “modding out in steps” can be equivalent to “modding out all at once,” provided the subspaces are nested appropriately.

5.2.2 When \(W_1 \subseteq W_2\): \((V/W_1)/(W_2/W_1)\)

If \(W_1 \subseteq W_2 \subseteq V\), then \[ (V/W_1)/(W_2/W_1) \cong V/W_2. \] At a conceptual level, \(V/W_1\) identifies vectors up to differences in \(W_1\), and then \(W_2/W_1\) identifies the remaining differences up to \(W_2\). The net effect is the same as identifying vectors differing by any element of \(W_2\) from the start.

6 Exact sequences viewpoint (optional linear-algebra framing)

6.1 Short exact sequences and quotients

A short exact sequence is an algebraic pattern describing how a quotient arises from a kernel. In the setting of vector spaces, a typical short exact sequence has the form \[ 0 \to W \to V \xrightarrow{\pi} V/W \to 0, \] where the injection identifies \(W\) with its image in \(V\), and the projection \(\pi\) is the canonical map whose kernel is exactly that copy of \(W\).

Exactness at \(V/W\) means the map onto the quotient is surjective, and exactness at \(V\) means the kernel is precisely the subspace we quotient by.

6.2 Connecting kernels, images, and quotients

6.2.1 Interpreting \(V/\ker T\) as \(\operatorname{im} T\)

For a linear transformation \(T:V\to U\), one can arrange an exact sequence \[ 0 \to \ker T \to V \to \operatorname{im}T \to 0, \] where the last map is the induced map on the quotient. Exactness captures the same relationship as the First Isomorphism Theorem: the quotient by the kernel yields a vector space naturally isomorphic to the image.

This viewpoint emphasizes that quotient spaces encode how much of \(V\) survives after applying \(T\).

7 Examples and computations

7.1 Quotient by a simple subspace in \(\mathbb{R}^n\)

Let \(V=\mathbb{R}^n\) and let \(W\) be spanned by some vectors, say \(W=\operatorname{span}(w_1,\dots,w_k)\). Two vectors \(x,y\in\mathbb{R}^n\) represent the same element of \(V/W\) exactly when \(x-y\) lies in that span.

To compute in practice, one chooses a basis extending \(w_1,\dots,w_k\) to a basis of \(\mathbb{R}^n\), then identifies quotient coordinates with the coefficients along the extra basis directions.

7.2 Quotient by a line or plane

In \(\mathbb{R}^2\), take \(W\) to be a line through the origin. Then \(V/W\) is one-dimensional: every coset corresponds to a direction perpendicular to \(W\) (in the sense of which components remain after removing the component along \(W\)).

In \(\mathbb{R}^3\), if \(W\) is a plane through the origin, the quotient \(V/W\) is one-dimensional as well. Elements of \(V/W\) correspond to “heights” along a normal direction to the plane, though the construction does not require selecting a metric or orthogonality—only a subspace.

7.3 Computing with different representatives

Suppose \(V=\mathbb{R}^2\) and \(W=\operatorname{span}((1,0))\). Then \((a,b)\) and \((a',b')\) lie in the same coset iff their difference is \((a-a',0)\), so this requires \(b=b'\). Thus the coset \( (a,b)+W\) can be represented by \((0,b)+W\).

This illustrates a common computational tactic: choose representatives that simplify the coset, often by removing the component in the direction of \(W\).

7.4 Quotient spaces in polynomial examples (e.g., mod a subspace/ideal context)

A frequent source of quotients appears when working with polynomials modulo constraints. While ideals are an algebraic structure rather than a plain subspace, the linear-algebra quotient idea underlies common constructions: forming a quotient identifies polynomials that differ by expressions regarded as “zero” under the imposed rule.

In such settings, quotient elements correspond to equivalence classes of polynomials, and computations reduce to working with remainders relative to the chosen constraints. The result is a smaller space that retains exactly the information not eliminated by the constraint.