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$A^{T}$ =In=$A^{T}$ 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 AT mean conceptually/geometrically (specifically when it does not equal the inverse of A, $A^{-1}$ )?