**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.4 Exercise 7.4.1**

Solution: In Exercise 7.2.6, we saw that $AE_{i,j}$ is the matrix whose j-th column is the i-th column of $A$, and all other columns are zero. Now let $A \in L_j$, and define $B \in M_n(R)$ to be the matrix whose i-th column is the j-th column of $A$ and all other entries are 0. Then $BE_{i,j} = A$, and hence $A \in M_n(R) E_{i,j}$. Now let $AE_{i,j} \in M_n(R) E_{i,j}$; clearly then all entries off of the j-th column of $AE_{i,j}$ are zero. Thus $AE_{i,j} \in L_j$.