Left ideals of $M_n(K)$ [duplicate]

Solution 1:

Let $I$ be an ideal in $M_n(K)$. Consider the set $V=e_1I$ where $e_1=(1,0,\dots,0) $. It is a $K$-vector space. Now you have to show that any matrix $M$ in $I$ has rows in $V$. Well, to check the $ i$th row, look at $ e_i I = e_1 P I $ where $ P$ is some permutation matrix that takes $ 1 $ to $ i $. Next, you have to show that for any $ v_1, \dots, v_n $ in $ V $, the matrix with rows $ v_i $ is in $ I $, but you can do something similar, for example using the fact that you know there are matrices $ M_i \in I $ with $ e_1 M_i = v_i $. Then if $ Q_i $ is the matrix with $ 1 $ in the $ (1,i) $ place and zeros everywhere else, then $ \sum Q_i M_i $ is the matrix with rows $ v_i $. There is probably an easier way to show this.