What's the meaning of the transpose? I don't understand the motivation of the transpose (or better yet, I haven't even seen one). It feels like just something pulled out of a hat. Thinking about it makes it seem  like a product of being able to write a matrix using either columns or rows 'first'. E.g., when we 'reflect down the diagonal' we are really keeping all the information of our old matrix, just changing rows to columns. 
This is much like how $f \text{  from} \{1,2,...,m-1,m\} \times \{1,2,...,n\} \text{ to some field } \mathbf{F}$ is an $m \times n$ matrix,
● if we switch the order of the product, so that we would define: $g \text{ is a map from } \{1,2,...,n\} \times \{1,2,...,m - 1, m\} \text{ to } \mathbf{F}$
● but allow $f(i,j) = g(j,i)$.
Then we somehow get a 'natural' map from the space of $m \times n$ matrices to $n \times m$ matrices.
Is that what transpose is - a 'natural' map for those spaces? What do I even mean by natural here (serious question, I'm not being mysterious)? If I had to guess, it is a linear bijective map? Is that close enough?
Moving away from this, does the transpose have any useful application in Euclidean geometry (other than orthogonal matrices being defined in terms of transposes?).
 A: In $\mathbb{R}^n$ with the standard inner product, $\langle x,y\rangle=\sum_i x_iy_i$, and for $A:\mathbb{R}^n\to\mathbb{R}^n$ linear, we have $\langle Ax,y\rangle=\langle x,A^ty\rangle$.  for instance, an orthogonal transformation $A$ preserves the inner product so that $\langle x,y\rangle=\langle Ax,Ay\rangle=\langle x,A^tAy\rangle$ and we find that $A^tA=I$.
A: 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.
A: 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). 
