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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.

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1 Answer 1

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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.

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