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

Let $$K$$ be a field and $$n \in \mathbb N$$. Show the following:

(i) Let $$V \subset K^n$$ be a subspace and $$I_V$$ the subset of $$M_n(K)$$ consisting of all the matrices whose rows belong to $$V$$. Prove that $$I_V$$ is a left ideal of $$M_n(K)$$.

(ii) Show that every left ideal of $$M_n(K)$$ is of the form defined in (i).

I think I could show (i). I am having problems with (ii)

For (i), take $$M \in M_n(K)$$ and $$N \in I_V$$. Let $$P=MN$$, I want to show that $$P \in I_V$$. Let $$P_i=(P_{i1} ... P_{in})$$ be the i-th row of $$P$$. Then, by definition of matrix multiplication, $$P_i=(\sum_{k=1}^n M_{ik}N_{k1} ... \sum_{k=1}^n M_{ik}N_{kn})$$ $$=\sum_{k=1}^n M_{ik} (N_{k1} ... N_{kn})$$ $$=M_{i1}(N_{11} ... N_{1n})+...+M_{in}(N_{n1} ... N_{nn})$$

This means that the $$i-th$$ row of $$P$$ is a linear combination of the rows of $$N$$. From here it follows $$P=MN \in I_V$$. This proves $$I_V$$ is a left ideal of $$M_n(K)$$.

I don't know how to show (ii), I would appreciate some help with that part.

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.