1) The set of mxn matrices with real elements, and real scalars is a vector space of dimension mxn.
2) The set of mxn matrices with complex elements and real scalars is a vector space of dimension 2xmxn:
Example: 1x2 matrix: [(a,b) (c,d)] with basis E1 = [(1,0) (0,0)], E2 = [(0,1) (0,0)], E3 = [(0,0) (1,0)], E4 = [(0,0) (0,1)].
[(a,b) (c,d)] = aE1 + bE2 + cE3 + dE4
3) The set of real mxn matrices with complex scalars is not a vector space because iA is not a real matrix.
4) The set of complex mxn matrices with complex scalars is easily defined by usual definitions of matrix addition , and scalar multiplication. This is also a vector space because addition and multiplication by a scalar are defined and closed:
Addition is obvious and aA is also a complex matrix
a(A+B) = aA + bB because Z1(Z2+Z3) = Z1xZ2 + Z1xZ3 for individual elements
A basis and dimension for this case has to be thought out.