What's the meaning of the transpose? [duplicate]

The transpose is closely related to dual spaces. A linear transformation $T:V\to W$ gives rise to a linear transformation $T^*:W^*\to V^*$ of the dual spaces. The corresponding matrix is the transpose of the original one, when you consider dual bases.

See http://en.wikipedia.org/wiki/Dual_space#Transpose_of_a_linear_map.

The transpose can be thought of as a generalization, or perhaps linearization, of the transpose of a binary relation (defined by $x R^T y \Leftrightarrow y R x$; for example, "is the parent of" is the transpose of "is the child of"). Indeed, it is possible to represent a relation between two sets $A, B$ as an $|A|$-by-$|B|$ matrix of $0$s and $1$s, and then the transpose of this matrix is the matrix of the transpose relation. If one thinks of a linear transformation between inner product spaces as a "linear relation," then its transpose is the "transpose linear relation" (which always exists, unlike the inverse).