The following is from this Wikipedia page:
Let $\mathbf{F}^{m\times n}$ denote the set of $m\times n$ matrices with entries in $\mathbf{F}$. Then $\mathbf{F}^{m\times n}$ is a vector space over $\mathbf{F}$. Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar). The zero vector is just the zero matrix. The dimension of $\mathbf{F}^{m\times n}$ is $mn$. One possible choice of basis is the matrices with a single entry equal to 1 and all other entries $0$.
My question is: Is there a notion of "four fundamental subspaces" in this case as well? If so, then what does it mean to be a "column space" of a vector space of matrices $\mathbf{F}^{m\times n}$? What about the "null space", "row space", and "left null space"?
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$\begingroup$''Column space'',''row space''and ''null space'' are defined for a Matrix, and are subspaces of the vector space in which the linear transformation represented by the matrix operates.
So, if you have a matrix $A\in M_{n\times n}(\mathbb{R})$ you can define these space as subspace of $\mathbb{R}^m$ or $\mathbb{R}^n$. But if you consider the matrix $A$ as an element of a vector space isomorphic to $\mathbb{R}^{mn}$ than, in this space the matrix becomes a vector and, for such vector the notions of ''Column space'',''row space'' etc.. has no meaning.
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