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

[matqkks]

What is the geometric interpretation of the transpose of a matrix? Is there any physical significance of the matrix transpose?

 

[Tinyboss]

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.