5.3 Transposes

  • Definition. Let \(\phi \in L(V,W)\) be a linear map of vector spaces. The transpose \(\phi ^T\) of \(\phi \) is the map \(\phi ^T:W^{*}\to V^{*}\) given by

    \begin{equation*} \phi ^T(\alpha ):=\alpha \circ \phi , \end{equation*}

    for all \(\alpha \in W^{*}\).

  • Lemma 5.13. \(\phi ^T:W^{*}\to V^{*}\) is also a linear map.

  • Proposition 5.14. Let \(V,W\) be finite-dimensional vector spaces and \(\phi \in L(V,W)\) with matrix \(A\in M_{m\times n}(\F )\) with respect to bases \(\lst {v}1n\) and \(\lst {w}1m\) of \(V\) and \(W\).

    Then \(\phi ^T\) has matrix \(A^T\) with respect to the dual bases \(\dlst {w}1m\) and \(\dlst {v}1n\) of \(W^{*}\) and \(V^{*}\).

  • Theorem 5.15. Let \(\phi \in L(V,W)\) be a linear map of vector spaces. Then

    • (1)  

      \begin{align*} \ker \phi &=\sol (\im \phi ^T)\\ \im \phi &\leq \sol (\ker \phi ^T) \end{align*} with equality if \(V,W\) are finite-dimensional.

    • (2)  

      \begin{align*} \ker \phi ^T&=\ann (\im \phi )\\ \im \phi ^T&\leq \ann (\ker \phi ) \end{align*} with equality if \(V,W\) are finite-dimensional.

  • Corollary 5.16. Let \(\phi \in L(V,W)\) be a linear map of finite-dimensional vector spaces. Then

    \begin{equation*} \rank \phi =\rank \phi ^T. \end{equation*}

  • Proposition 5.17. Let \(\phi \in L(V,W)\) be a linear map of finite-dimensional vector spaces. Then

    • (1) \(\phi \) injects if and only if \(\phi ^T\) surjects.

    • (2) \(\phi ^T\) injects if and only if \(\phi \) surjects.