2.3 Quotients

Let \(U\leq V\). We construct a new vector space from \(U\) and \(V\) which is an “abstract complement” to \(U\). The elements of this vector space are equivalence classes for the following equivalence relation:

  • Definition. Let \(U\leq V\). Say that \(v,w\in V\) are congruent modulo \(U\) if \(v-w\in U\). In this case, we write \(v\equiv w\mod U\).

  • Warning. This is emphatically not the relation of congruence modulo an integer \(n\) that you studied in Algebra 1A: here the relation is between vectors in a vector space. However, both notions of congruence are examples of a general construction in group theory.

  • Lemma 2.10. Congruence modulo \(U\) is an equivalence relation.

  • Proof. Exercise3! □

3 This is question 3 on exercise sheet 2.

Thus each \(v\in V\) lies in exactly one equivalence class \([v]\sub V\).

What do these equivalence classes look like? Note that \(w\equiv v\mod U\) if and only if \(w-v\in U\) or, equivalently, \(w=v+u\), for some \(u\in U\).

  • Definition. For \(v\in V\), \(U\leq V\), the set \(v+U:=\set {v+u\st u\in U}\sub V\) is called a coset of \(U\) and \(v\) is called a coset representative of \(v+U\).

We conclude that the equivalence class of \(v\) modulo \(U\) is the coset \(v+U\).

(A subspace and one of its cosets.)

Figure 2.4: A subspace \(U\leq \R ^{2}\) and a coset \(v+U\).

  • Remark. In geometry, cosets of vector subspaces are called affine subspaces. Examples include lines in \(\R ^2\) and lines and planes in \(\R ^3\) irrespective of whether they contain zero (as vector subspaces must).

  • Example. Fibres of a linear map: let \(\phi :V\to W\) be a linear map and let \(w\in \im \phi \). Then the fibre of \(\phi \) over \(w\) is defined by:

    \begin{equation*} \phi ^{-1}\set {w}:=\set {v\in V\st \phi (v)=w}. \end{equation*}

    Unless \(w=0\), this is not a linear subspace but notice that \(v,v'\) are in the same fibre if and only if \(\phi (v)=\phi (v')\), or, equivalently, \(\phi (v-v')=0\) or \(v-v'\in \ker \phi \). We conclude that the fibres of \(\phi \) are exactly the cosets of \(\ker \phi \):

    \begin{equation*} \phi ^{-1}\set {w}=v+\ker \phi , \end{equation*}

    for any \(v\in \phi ^{-1}\set {w}\).

    We shall see below that any coset arises this way for a suitable \(\phi \).

  • Definition. Let \(U\leq V\). The quotient space \(V/U\) of \(V\) by \(U\) is the set \(V/U\), pronounced “\(V\) mod \(U\)”, of cosets of \(U\):

    \begin{equation*} V/U:=\set {v+U\st v\in V}. \end{equation*}

    This is a subset of the power set4 \(\mathcal {P}(V)\) of \(V\).

    The quotient map \(q:V\to V/U\) is defined by

    \begin{equation*} q(v)=v+U. \end{equation*}

4 Recall from Algebra 1A that the power set of a set \(A\) is the set of all subsets of \(A\).

The quotient map \(q\) will be important to us. It has two key properties:

  • (1) \(q\) is surjective.

  • (2) \(q(v)=q(v')\) if and only if \(v\equiv v'\mod U\), that is, \(v-v'\in U\).

We can add and scalar multiply cosets to make \(V/U\) into a vector space and \(q\) into a linear map:

  • Theorem 2.11. Let \(U\leq V\). Then, for \(v,w\in V\), \(\lambda \in \F \),

    \begin{align*} (v+U)+(w+U)&:=(v+w)+U\\ \lambda (v+U)&:=(\lambda v)+U \end{align*} give well-defined operations of addition and scalar multiplication on \(V/U\) with respect to which \(V/U\) is a vector space and \(q:V\to V/U\) is a linear map.

    Moreover, \(\ker q=U\) and \(\im q=V/U\).

  • Proof. We phrase everything in terms of \(q\) to keep the notation under control. Since \(q\) surjects, we lose nothing by doing this: any element of \(V/U\) is of the form \(q(v)\) for some \(v\in V\).

    With this understood, the proposed addition and scalar multiplication in \(V/U\) read

    \begin{align*} q(v)+q(w)&:=q(v+w)\\\lambda q(v)&:=q(\lambda v) \end{align*} so that \(q\) is certainly linear so long as these operations make sense. Here the issue is that if \(q(v)=q(v')\) and \(q(w)=q(w')\), we must show that

    \begin{equation} \label {eq:5} q(v+w)=q(v'+w'),\qquad q(\lambda v)=q(\lambda v'). \end{equation}

    However, in this case, we have \(v-v'\in U\) and \(w-w'\in U\) so that

    \begin{gather*} (v+w)-(v'+w')=(v-v')+(w-w')\in U\\ \lambda v-\lambda v'=\lambda (v-v')\in U, \end{gather*} since \(U\) is a subspace, and this establishes (2.1).

    As for the vector space axioms, these follow from those of \(V\). For example:

    \begin{equation*} q(v)+q(w)=q(v+w)=q(w+v)=q(w)+q(v). \end{equation*}

    Here the first and third equalities are the definition of addition in \(V/U\) and the middle one comes from commutativity of addition in \(V\). The zero element is \(q(0)=0+U=U\) while the additive inverse of \(q(v)\) is \(q(-v)\).

    The linearity of \(q\) comes straight from how we defined our addition and scalar multiplication while \(v\in \ker q\) if and only if \(q(v)=q(0)\) if and only if \(v=v-0\in U\) so that \(\ker q=U\). □

(Illustration of the quotient map.)

Figure 2.5: The quotient map \(q\).

  • Corollary 2.12. Let \(U\leq V\). If \(V\) is finite-dimensional then so is \(V/U\) and

    \begin{equation*} \dim V/U=\dim V-\dim U. \end{equation*}

  • Proof. Apply rank-nullity to \(q\) using \(\ker q=U\) and \(\im q=V/U\). □

  • Remark. Theorem 2.11 shows that:

    • (1) Any \(U\leq V\) is the kernel of a linear map.

    • (2) Any coset \(v+U\) is the fibre of a linear map: indeed

      \begin{equation*} v+U=q^{-1}\set {q(v)}. \end{equation*}

  • Commentary. Many people find the quotient space \(V/U\) difficult to think about: its elements are (special) subsets of \(V\) and this can be confusing.

    An alternative, perhaps better way, to proceed is to concentrate instead on the properties of \(V/U\) in much that same way that, in Analysis, we deal with real numbers via the axioms of a complete ordered field without worrying too much what a real number actually is!

    From this point of view, the quotient \(V/U\) of \(V\) by \(U\) is a vector space along with a linear map \(q:V\to V/U\) such that

    • • \(q\) surjects;

    • • \(\ker q=U\)

    and this is really all you need to know!

    The content of Theorem 2.11, from this perspective, is simply that quotients exist!

  • Theorem 2.13 (First Isomorphism Theorem). Let \(\phi :V\to W\) be a linear map of vector spaces.

    Then \(V/\ker \phi \cong \im \phi \).

    In fact, define \(\bar {\phi }:V/\ker \phi \to \im \phi \) by

    \begin{equation*} \bar {\phi }(q(v))=\phi (v), \end{equation*}

    where \(q:V\to V/\ker \phi \) is the quotient map.

    Then \(\bar {\phi }\) is a well-defined linear isomorphism.

  • Proof. First we show that \(\bar {\phi }\) is well-defined: \(q(v)=q(v')\) if and only if \(v-v'\in \ker \phi \) if and only if \(\phi (v-v')=0\), or, equivalently, \(\phi (v)=\phi (v')\). We also get a bit more: \(\bar {\phi }\) injects since if \(\bar {\phi }(q(v))=\bar {\phi }(q(v'))\) then \(\phi (v)=\phi (v')\) which implies that \(q(v)=q(v')\).

    To see that \(\bar {\phi }\) is linear, we compute using the linearity of \(q\) and \(\phi \):

    \begin{equation*} \bar {\phi }(q(v_1)+\lambda q(v_2))= \bar {\phi }(q(v_1+\lambda v_2))= \phi (v_1+\lambda v_2)= \phi (v_1)+\lambda \phi (v_2)= \bar {\phi }(q(v_1))+\lambda \bar {\phi }(q(v_2)), \end{equation*}

    for \(v_1,v_2\in V\), \(\lambda \in \F \).

    It remains to show that \(\bar {\phi }\) is surjective: but if \(w\in \im \phi \), then \(w=\phi (v)=\bar {\phi }(q(v))\), for some \(v\in V\), and we are done. □

  • Remarks.

    • (1) Let \(q:V\to V/\ker \phi \) be the quotient map and \(i:\im \phi \to W\) the inclusion. Then the First Isomorphism Theorem shows that we may write \(\phi \) as the composition \(i\circ \bar {\phi }\circ q\) of a quotient map, an isomorphism and an inclusion.

    • (2) This whole story of cosets, quotients and the First Isomorphism Theorem has versions in many other contexts such as group theory and ring theory (see MA22017).