What's the intuition of the transpose of a matrix? I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition of transpose. Suppose $A_{m \times n}$, and A transform a vector from $\Bbb R^n$ to $\Bbb R^m$. But $A^T$ transform a vector from $\Bbb R^m$ to $\Bbb R^n$. What's the relationship between them? Could anyone please explain the relationship between $A^T$,$A$,the inner product and symmetric matrix. I think there would be a intuition explaination.
 A: Well, $A^T$ is the adjoint matrix of $A$ with respect to the ordinary inner products, i.e. $A^T$ is the only linear mapping $B$ such that
$$\langle Av,w\rangle = \langle v,Bw\rangle$$
for all $v\in\Bbb R^n$ and $w\in\Bbb R^m$. You can easily see it if you verify it on the standard bases, noting that $\langle u,e_i\rangle$ gives the $i$th coordinate of $u$.
A: Good geometric ilustration of transpose is if we take as linear operator rotation matrix R.
In this case is easy to see that < Rv, w> = < v, $R^T$ w> as we have 2 opportunities to change an angle between vector v and w to the same value.
One opportunity is to rotate v what   Rv operation does, the second one to rotate w in reverse direction what $R^T$  does ( transpose of R is inverse of R).
A: One aspect of this to consider is that the transpose lets you do the same thing in different ways. The regular matrix gets multiplied by a column on the right to give your answer as a column. The transpose gets multiplied by a row on the left to give a row as the answer. 
Is one better than the other? Not really, they're equivalent.
