Let us work over the field $\mathbb{C}$ say. Consider the space $M_{m \times 2}(\mathbb{C})$ of $m \times 2$ matrices over $\mathbb{C}$, as a vector space over $\mathbb{C}$. Let $V$ be some subspace of dimension $2n$ for some $n \leq m$. Since $V$ is a vector space of dimension $2n$, there exist $2n$ basis vectors (in this case, $m \times 2$ matrices) that generate $V$ over $\mathbb{C}$. This is all basic.

What I am interested in is whether this subspace $V$ can be expressed as the image of an $m \times n$ matrix over $\mathbb{C}$. That is, does there exist a matrix $A$ such that $$V=\{ A\boldsymbol{b} : \boldsymbol{b} \in M_{n \times 2}(\mathbb{C})\}$$

For instance, it is possible that all matrices in $V$ have rank $2$ (this is possible), but for a matrix $\boldsymbol{b}$ of rank $1$, the product $A\boldsymbol{b}$ would also have rank $1$, so in this case $V$ cannot be expressed this way? This is my intuition, but I am unable to formalize this to say more about what makes a space $V$ expressible this way. And if not, is there some "next best thing"? This seems like an elementary question, but I'm a bit confused when it comes to treating matrices as vectors.

More generally, when can a space of $m \times k$ matrices of dimension $kn$ be expressed as the image of an $m \times n$ matrix?


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