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Let $A \in \mathcal{M}_{n \times n}$ be a square matrix of size $n$ with rank $k \in [0,n]$. Define $$ L = \{B \in \mathcal{M}_{n \times n} : BA = 0 \}, \\ R = \{C \in \mathcal{M}_{n \times n} : AC = 0 \}. $$ What is the dimension of $L$ and $R$?

I know $L$ and $R$ are linear subspaces.
By the dimension theorem $$ \text{nullity}(A)=n-k, $$
so, I would be tempt to say $$ \dim(R)=\text{nullity}(A)=n-k. $$ but I'm not sure, because $A$ acts on $\mathbb R^n$, while the element of $R$ acts on $\mathcal{M}_{n \times n}$.
For $L$, I have no idea.

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2 Answers 2

up vote 2 down vote accepted

Observe that the if $C\in R$, then the columns of $C$ are in $\ker A$. Let $v_1,...,v_{n-k}$ be a basis for $\ker A$. Then the $(n-k)n$ matrices with zeroes in all clumns and one of the $v_1,...,v_{n-k}$ in one of the columns form a basis for $R$. Hence $\dim R=n(n-k)$

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Hint:
$C\in R\iff $ each column of $C$ is an element of $\ker (A)$.
For $L$ is the same using the matrix $A^t$.

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