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Good Afternnon

If $f$ is a linear transformation mapping then if the transformation matrix of linear transformation mapping has been defined and its transpose properties are known as well.

Why, what I have written below, is called a definition rather than a proposition?? Is it possible to deduce it from the matrices?

$E,F$ two finite vector-spaces.

$\psi\in \mathcal{L}(E,F)$. The transpose of $\psi$ is the linear map :

$\begin{array}l ^t\psi :&F^*&\to E^*\\&\phi&\to \phi\circ\psi\end{array}$

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Because the transpose of a matrix is one thing and the tranpose of a linear map is a distinct concept.

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