Question

Suppose $V, W$ are finite-dimensional inner product spaces with dual space $V^, W^$. Let $L: V \rightarrow W$ be a linear function, and let $\bar{L}^: W^ \rightarrow V^$ denote the dual linear function, as in Exercise 7.2 .30 (without the tilde), while $L^: W \rightarrow V$ denotes its adjoint. (As noted above, the same notation denotes two mathematically different objects.) Prove that if we identify $V^* \simeq V$ and $W^* \simeq W$ using the linear isomorphism in Exercise 7.1.62, then the dual and adjoint functions are identified $\bar{L}^* \simeq L^*$, thus reconciling the unfortunate clash in notation. In particular, this includes the two possible interpretations of the transpose of a matrix.

    Suppose $V, W$ are finite-dimensional inner product spaces with dual space $V^*, W^*$. Let $L: V \rightarrow W$ be a linear function, and let $\bar{L}^*: W^* \rightarrow V^*$ denote the dual linear function, as in Exercise 7.2 .30 (without the tilde), while $L^*: W \rightarrow V$ denotes its adjoint. (As noted above, the same notation denotes two mathematically different objects.) Prove that if we identify $V^* \simeq V$ and $W^* \simeq W$ using the linear isomorphism in Exercise 7.1.62, then the dual and adjoint functions are identified $\bar{L}^* \simeq L^*$, thus reconciling the unfortunate clash in notation. In particular, this includes the two possible interpretations of the transpose of a matrix.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 7, Problem 10

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Step 1

- $V^*, W^*$ are the dual spaces of $V$ and $W$, respectively. - $L: V \rightarrow W$ is a linear function. - $\bar{L}^*: W^* \rightarrow V^*$ is the dual linear function of $L$, defined by $\bar{L}^*(\phi) = \phi \circ L$ for $\phi \in W^*$. - $L^*: W \rightarrow Show more…

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Suppose $V, W$ are finite-dimensional inner product spaces with dual space $V^*, W^*$. Let $L: V \rightarrow W$ be a linear function, and let $\bar{L}^*: W^* \rightarrow V^*$ denote the dual linear function, as in Exercise 7.2 .30 (without the tilde), while $L^*: W \rightarrow V$ denotes its adjoint. (As noted above, the same notation denotes two mathematically different objects.) Prove that if we identify $V^* \simeq V$ and $W^* \simeq W$ using the linear isomorphism in Exercise 7.1.62, then the dual and adjoint functions are identified $\bar{L}^* \simeq L^*$, thus reconciling the unfortunate clash in notation. In particular, this includes the two possible interpretations of the transpose of a matrix.

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