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I'm struggling to understand what the transpose of a linear transformation represents. My textbook's motivation for this wasn't very helpful. All it did was ask, "Is there a linear transformation $U$ associated with $T: V \to W$ in a natural way such that $U$ may be represented by $([T]_{\beta}^{\gamma})^t$?" I guess this is an interesting question but the problem is that $U$, to me, is certainly not associated in a "natural" way. Defining $U$ as $T^t: W^* \to V^*$ by $T^t(g) = gT$ for all $g \in W^*$ is not natural in my opinion.

Can anybody help me by explaining what this function represents and how its useful in math? (Also note that my knowledge of math isn't very extensive, so please try to not be too technical)

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  • Does it have to represent anything natural? I think it's used mostly to do the dot product of 2 vectors (or vector sets, which matrices may represent). To do that we need to transpose one of the vectors (or matrices). – Stanislav Bashkyrtsev Aug 08 '23 at 11:25

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Transpose of a linear map $T: V \to W$ is an induced pull-back of linear functionals $g \in W^* $: we want them to act back on the original $V$ somehow. So essentially we're looking for something we denote as a decorated symbol $T^t: W^* \to V^*.$

It is indeed possible and trivial if we actually make use of $T$ to do its job: move elements between $V$ and $W$. Hence the notation $T^t$ and defining formula: $T^t(g) = g\circ T,$ giving us a functional acting on $V$ now.

rych
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