Suppose I have $T: V \to W$. Then $T$ defines a map $T': W' \to V'$ that some sources (e.g. Axler) call the dual map of $T$. And $T$ defines a map $T^t: W' \to V'$ that some sources (e.g. Hoffman and Kunze) call the transpose of $T$. My question is: are these synonyms?
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1$\begingroup$ Dual map, transpose map, adjoint map(though this has other meanings as well) all refer to the same thing: the function from $W'\to V'$ such that $f\mapsto f\circ T$. $\endgroup$– peek-a-booMar 8, 2022 at 21:48
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$\begingroup$ Amazing. That is exactly what I was hoping you would say. Thanks so much! $\endgroup$– 1Teaches2LearnMar 8, 2022 at 21:48
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$\begingroup$ Just elaborating on my previous comment. The reason we call $T'$ the dual map because it maps between the dual spaces and the word 'dual' is used because $V,V'$ are kind of similar but then again not really, same with $W,W'$ and $T,T'$. The reason why we call $T'=T^t$ the transposed map is because if you take a basis $\beta$ of $V$, $\gamma$ of $W$ and the unique dual bases $\beta',\gamma'$ of $V',W'$ then the matrix representations are related by transpose: $[T^t]_{\gamma'}^{\beta'}=\left([T]_{\beta}^{\gamma}\right)^t$. $\endgroup$– peek-a-booMar 8, 2022 at 21:56
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$\begingroup$ Thanks. That's sort of exactly why I was asking. I'm trying to understand the relationship between "dual map" "transpose" and "adjoint". And what I've found, in short is, "dual map" = "transpose", and "dual map" = "adjoint" if you have an inner product so that you can identify the dual space of V with V itself through Reisz representation of linear functionals as vectors. (Although, it seems that also, from a higher level perspective which I don't yet fully understand, these distinctions begin to break down or seem less relevant, hence why the terminology at the lower level can seem murky.) $\endgroup$– 1Teaches2LearnMar 8, 2022 at 22:40
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1$\begingroup$ Theorem 3.114 of my book Linear Algebra Done Right (third edition) states that the matrix of $T'$ with respect to the dual bases is the transpose of the matrix of $T$. $\endgroup$– Sheldon AxlerMar 9, 2022 at 3:21
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