2.3 Quotients

  • 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 \(v\equiv w\mod U\).

  • Lemma 2.12. Congruence modulo \(U\) is an equivalence relation.

  • 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 \(v+U\).

  • 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 set3 \(\mathcal {P}(V)\) of \(V\).

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

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

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

  • Theorem 2.13. 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 \(\im q=V/U\).

  • Corollary 2.14. 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*}

  • Theorem 2.15 (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.