# How to approach vector space of matrices?

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"?

• Sure, these subspaces exist. They are just spanned by matrices instead of row/column vectors. – Matthew Leingang Oct 4 '16 at 18:49
• What do you mean by "four fundamental subspaces"? I cannot find a definition in your question. – Dietrich Burde Oct 4 '16 at 18:52
• For vector space comprising of "vectors", we find the basis vectors for the space spanned by those vectors and stack them in a matrix $A$. From $A$, we can define four subspaces namely column space, row space, null space, and left null space. How do we do this when our elements in space are matrices instead of vectors – NAASI Oct 4 '16 at 18:57
• – Rodrigo de Azevedo Oct 4 '16 at 19:29
• I see. And I thought, "four" was just a bit arbitrary. So the "eight fundamental subspaces" would be these plus $\ker(A^2),im(A^2),coker (A^2),coim(A^2)$ ? – Dietrich Burde Oct 4 '16 at 19:58

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.
• does that mean we can have a space $\mathcal{S}$ whose elements are "matrices" but we cannot an an equivalent of column space, row space, etc in this context – NAASI Oct 4 '16 at 19:31
• Yes we can, but for matrices that represents linear transformations of the matrices that are elements of $S$. Each element of $S$ have its proper row, column etc spaces, but these are not subspaces of $S$. – Emilio Novati Oct 4 '16 at 19:35