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Let $V$ be the vector space of all $m \times n$ matrices over some field $\Bbb F$.

What does this intuitively mean?

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It means that you can add two $m\times n$ matrices, and that you can multiply a $m\times n$ matrix with an element of $\mathbb F$ (a "scalar"). –  Rahul Narain Feb 3 at 5:35
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A vector space is defined a list of axioms. Anything satisfying those axioms is a vector space. With the standard operations on matrices the set you describe satisfies the axioms of a vector space. This is formal meaning it being a vector space.

Intuitively it means that the for certain aspects of matrix theory one can use intuition from any other vector space. Some vector spaces, like $\mathbb R^n$ carry a very rich geometric intuition and so this intuition can be transferred to the vector space of matrices.

This is the common approach of modern mathematics. Instead of defining what some gadget is we define what we can do with the gadgets. After all, who cares what something is made of and precisely how it does whatever it is that it does? All we need to care about is what we can actually do with it. What happens under the hood is irrelevant. So, both matrices and elements in $\mathbb R^n$ thought of as arrows or something exhibit similar properties when one takes a step away from the particularities of the objects and looks at what can be done with the objects. This is a very powerful technique used all over mathematics.

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When I saw your question the first thing that came to mind for “intuitive meaning” is geometric/spatial visualization; vector space as actual space. In this particular case, there is no simple answer because $V$ has $mn$ dimensions, which in the simplest nontrivial case is $4$. But we can try something anyway.

Everything I describe below is going to be for the case where $m=2, n=2$, and also $\mathbb{F} = \mathbb{R}$.

As you may know (if you don't, this isn't going to help), a $2×2$ matrix over $\mathbb{R}$, $\big[\begin{smallmatrix} a & c \\ b & d \end{smallmatrix}\big],$ can be employed as a linear transformation of 2D space, and in that case the matrix's columns are a set of basis vectors. So, visualize those basis vectors; in particular, visualize them as points $(a,b)$ and $(c,d)$ offset from some origin (and, if you like, include some input-to-the-transformation shape transformed to match).

enter image description here

The space $V$ consists of all possible pairs of locations of those two points; that is, that pair of vectors is a vector in $V$.

From the “four-dimensional visualization” perspective, a couple of points:

  • Moving one point while holding the other fixed moves in a 2D slice (a plane, and not a hyperplane) of the 4D space.
  • The plane on which both of the vectors are displayed above could be interpreted as a isometric projection of four-dimensional $V$ onto 2 dimensions — but in that case, the projection of a given vector is $(a+c, b+d)$, not the pair of vectors but their sum.

Generalizations: When $m > 2$, the column vectors are in an $m$-dimensional space (replace the coordinate plane depicted above). When $n > 2$, there are $n$ of them (rather than just the $2$, red and blue, depicted above).

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