5.2 Solution sets and annihilators

  • Definition. Let \(E\leq V^{*}\). The solution set of \(E\) is

    \begin{equation*} \sol E:=\set {v\in V\st \text {$\alpha (v)=0$, for all $\alpha \in E$}}=\bigcap _{\alpha \in E}\ker \alpha \leq V. \end{equation*}

  • Proposition 5.6. If \(V\) is finite-dimensional and \(E\leq V^{*}\) then

    \begin{equation*} \dim \sol E=\dim V-\dim E. \end{equation*}

    We say that \(E\) and \(\sol E\) have complementary dimension.

  • Corollary 5.7. Let \(V\) have dimension \(n\) and suppose that \(\lst {\alpha }1n\in V^{*}\) are such that

    \begin{equation*} \bigcap _{i=1}^n\ker \alpha _i=\set 0. \end{equation*}

    Then \(\lst \alpha 1n\) is a basis of \(V^{*}\).

  • Proposition 5.8. Let \(E,F\leq V^{*}\). Then

    • (1) If \(E\leq F\) then \(\sol F\leq \sol E\).

    • (2) \(\sol \) swaps sums and intersections:

      \begin{align*} \sol (E+F)&=(\sol E)\cap (\sol F)\\ (\sol E)+(\sol F)&\leq \sol (E\cap F) \end{align*} with equality if \(V\) is finite-dimensional.

  • Definition. Let \(U\leq V\). The annihilator of \(U\), denoted \(\ann U\) or \(U^{\circ }\), is given by:

    \begin{equation*} \ann U:=\set {\alpha \in V^{*}\st \alpha _{|U}=0}=\set {\alpha \in V^{*}\st \text {$\alpha (u)=0$, for all $u\in U$ }}. \end{equation*}

  • Proposition 5.9. Let \(V\) be finite-dimensional and \(U\leq V\). Then

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

  • Proposition 5.10. Let \(U,W\leq V\). Then

    • (1) If \(U\leq W\) then \(\ann W\leq \ann U\).

    • (2)  

      \begin{align*} \ann (U+W)&=(\ann U)\cap (\ann W)\\ (\ann U)+(\ann W)&\leq \ann (U\cap W) \end{align*} with equality if \(V\) is finite-dimensional.

  • Lemma 5.11. Let \(U\leq V\) and \(E\leq V^{*}\) then \(U\leq \sol E\) if and only if \(E\leq \ann U\).

  • Theorem 5.12. Let \(U\leq V\) and \(E\leq V^{*}\). Then

    \begin{align*} U&\leq \sol (\ann U)\\ E&\leq \ann (\sol E), \end{align*} with equality if \(V\) is finite-dimensional.