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Unfortunately, VectorSpace seems only to consider column vectors, not matrix as elements. So i do not know whether it is possible out of the box. A possible workaround is to "unfold" your matrix to a vector, and then refold result vector to get a matrix back.

Here is an example:

sage: p = 5
sage: F = GF(p)
sage: MM = MatrixSpace(F,2,3) ; MM
Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 5

sage: M1 = MM.random_element() ; M1
[2 0 0]
[1 4 3]

sage: M2 = MM.random_element() ; M2
[2 3 4]
[4 0 3]

sage: V = VectorSpace(F,MM.dimension()) ; V
Vector space of dimension 6 over Finite Field of size 5

sage: def unfold(M):
....:     return V(M.list())

sage: def refold(UM):
....:     return MM(UM.list())

sage: UM1 = unfold(M1)   
sage: UM2 = unfold(M2)
sage: refold(UM1) == M1
True

sage: sp = V.span([UM1,UM2]) ; sp

UM1 + UM2 in sp

sage: UM1 + UM2 in sp
True
sage: UM3 = sp.random_element()
sage: M3 = refold(UM3) ; M3
[2 4 2]
[0 2 3]

sage: M3 in MM
True

Unfortunately, VectorSpace seems only to consider column vectors, not matrix matrix, as elements. So i do not know whether it is possible out of the box. A possible workaround is to "unfold" your matrix to a vector, do your computations in the span, and then refold result vector to get a matrix back.

Here is an example:

sage: p = 5
sage: F = GF(p)
sage: MM = MatrixSpace(F,2,3) ; MM
Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 5

sage: M1 = MM.random_element() ; M1
[2 0 0]
[1 4 3]

sage: M2 = MM.random_element() ; M2
[2 3 4]
[4 0 3]

sage: V = VectorSpace(F,MM.dimension()) VectorSpace(F, MM.dimension()) ; V
Vector space of dimension 6 over Finite Field of size 5

sage: def unfold(M):
....:     return V(M.list())

sage: def refold(UM):
....:     return MM(UM.list())

sage: UM1 = unfold(M1)   
unfold(M1)
sage: UM2 = unfold(M2)
sage: refold(UM1) == M1
True

sage: sp = V.span([UM1,UM2]) V.span([UM1, UM2]) ; sp

UM1 + UM2 in sp
Vector space of degree 6 and dimension 2 over Finite Field of size 5
Basis matrix:
[1 0 0 3 2 4]
[0 1 3 1 2 0]

sage: UM1 + UM2 in sp
True
sage: UM3 = sp.random_element()
sage: M3 = refold(UM3) ; M3
[2 4 2]
[0 2 3]

sage: M3 in MM
True

Unfortunately, VectorSpace VectorSpace seems only to consider column row vectors, not matrix, matrices, as elements. So i do not know whether it is possible out of the box. A possible workaround is to "unfold" your matrix to a vector, matrices to vectors, do your computations in the span, and then refold result vector to get a matrix back.back. Folding and unfolding being isomorphisms of vector spaces, you should be safe with your computations.

Here is an example:

sage: p = 5
sage: F = GF(p)
sage: MM = MatrixSpace(F,2,3) ; MM
Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 5

sage: M1 = MM.random_element() ; M1
[2 0 0]
[1 4 3]

sage: M2 = MM.random_element() ; M2
[2 3 4]
[4 0 3]

sage: V = VectorSpace(F, MM.dimension()) ; V
Vector space of dimension 6 over Finite Field of size 5

sage: def unfold(M):
....:     return V(M.list())

sage: def refold(UM):
....:     return MM(UM.list())

sage: UM1 = unfold(M1)
sage: UM2 = unfold(M2)
sage: refold(UM1) == M1
True

sage: sp = V.span([UM1, UM2]) ; sp
Vector space of degree 6 and dimension 2 over Finite Field of size 5
Basis matrix:
[1 0 0 3 2 4]
[0 1 3 1 2 0]

sage: UM1 + UM2 in sp
True
sage: UM3 = sp.random_element()
sage: M3 = refold(UM3) ; M3
[2 4 2]
[0 2 3]

sage: M3 in MM
True