Representation of linear transformation of matrices The transformation of any matrix $A$ that switches two rows (or two columns) in $A$ can be represented as a product between $A$ and a transformation matrix.
The question is: what kind of linear transformations between (linear spaces of) matrices can be represented in a similar way? In particular, what is the answer for the transposition of matrices and for the rotation of matrices ($90$ degrees counterclockwise)?
Also, any good reference on the subject is welcomed.
[I assume it has to do with the dimension of the kernel of the transformation... but this a way of saying that I clearly need help, since my linear algebra is a little rusty]
 A: The corollary on page 20 of Linear Algebra by Hoffman and Kunze partially answers your first question: Let $A$ and $B$ be $m \times n$ matrices over a field $\Bbb F.$ Then $B$ is row-equivalent to $A$ if and only if $B = PA,$ where $P$ is a product of $m \times m$ elementary matrices.  We get the full answer if we allow some rows of $P$ to be zero: The kinds of linear transformations that can be represented by left multiplication are those such that $B$ is row-equivalent to $A$ but possibly with some zero rows in $B.$ Right multiplication gives an analogous result for columns.
Transposed matrices are not necessarily row-equivalent, so no such matrix exists (but see https://math.stackexchange.com/q/1143642).
Rotation of a vector counterclockwise in $\Bbb R^2$ by the angle $\theta$ is given by left multiplication by
$$\begin{bmatrix}
 \cos \theta & \sin \theta\\
-\sin \theta & \cos \theta\end{bmatrix}.$$
For $\theta = 90^\circ,$ the matrix is
$$\begin{bmatrix}
 0 & 1\\
-1 & 0\end{bmatrix}.$$
In $\Bbb R^3,$ see equation (20) in "Three-Dimensional Rotation Matrices."
