Finding the $3 \times 3$ transformation matrix with respect to standard basis for $A \mapsto A+A^T$

Question

$X=\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix} \mapsto X'=\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix} +\begin{bmatrix} x_{11} & x_{21} & x_{31} \\ x_{12} & x_{22} & x_{32} \\ x_{13} & x_{23} & x_{33} \end{bmatrix} = \begin{bmatrix} x_{11}+x_{11} & x_{12}+x_{21} & x_{13}+x_{31} \\ x_{21}+x_{12} & x_{22}+x_{22} & x_{23}+x_{32} \\ x_{31}+x_{13} & x_{32}+x_{23} & x_{33}+x_{33} \end{bmatrix} $

$L \ X = X'$
$L=?$

Answer 1

Hints:

1. Which vector space is the domain of the given function $L$? How many dimensional? What's the standard basis in this vector space?
2. Which vector space is the range of $L$?
3. Calculate each $L(E_k)$ in terms of the $(E_k)_k$ where $(E_k)_k$ is the standard basis from 1.