Transpose of a Matrix
The transpose $A^{T}$ of a matrix A is mechanically obtained by writing the rows of A as columns of $A^{T}$ and, simultaneously, writing the columns of A as the rows of $A^{T}$. When A is an orthogonal matrix, A$A^{T}$ =$I_{n}$=$A^{T}$A, so $A^{T}$ must be the inverse matrix of A ($A^{T}$ =$A^{-1}$). If the A matrix represents some linear transformation T: V --> W, what does $A^{T}$ mean conceptually/geometrically (specifically when it does not equal the inverse of A, $A^{-1}$)?