What is the relationship between a matrix and its transpose in linear algebra?

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The relationship between a matrix and its transpose in linear algebra is significant in both geometric and physical contexts. The transpose of a matrix represents a linear map between dual spaces, indicating how linear functionals interact with vector spaces. Geometrically, the transpose can be interpreted as reflecting transformations across a diagonal in a coordinate system. While the mathematical properties are clear, the physical significance remains less intuitive and requires deeper exploration. Understanding this relationship enhances comprehension of linear transformations and their applications in various fields.
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What is the geometric interpretation of the transpose of a matrix? Is there any physical significance of the matrix transpose?
 
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matqkks said:
What is the geometric interpretation of the transpose of a matrix? Is there any physical significance of the matrix transpose?

In a finite-dimensional vector space, the dual space (the space of linear functionals) is isomorphic to the vector space, and if A is the matrix for a linear map from vector spaces V to W (with bases X and Y), then there is an induced linear map between the dual spaces W* to V* (w.r.t. the dual bases X* and Y*), whose matrix is the transpose of A. But I don't know what the physical or geometric intuition is, if any, for this relationship.
 

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