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?).