What is the physical meaning of the transpose of a matrix? Geometrically, does it show only reflection or does it have other geometrical significance too?
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Do you know what a basis of a vector space is? A linear transformation between vector spaces? And what is a linear functional on a vector space? To each linear transformation $T: \mathbb R^n \rightarrow \mathbb R^m$, which can be represented by a matrix $A$ with the standard basis, the transpose $A^T$ represents the "dual transformation" $T^{\ast}: (\mathbb R^m)^{\ast} \rightarrow (\mathbb R^n)^{\ast}$ with respect to the dual basis. – D_S Dec 07 '19 at 03:56

What do you consider to be the physical meaning of the original, untransposed matrix? – amd Dec 07 '19 at 08:06