I can't understand what Roam said at all since the words "independent" and "spanning" apply to sets of vectors, not individual vectors or matrices.
millerst, you probably know that a basis for, say, , the set of quadruples, (a, b, c, d), is {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} since we can write (a, b, c, d)= (a, 0, 0, 0)+ (0, b, 0, 0)+ (0, 0, c, 0)+ (0, 0, 0, d)= a(1, 0, 0, 0)+ b(0, 1, 0, 0)+ c(0, 0, 1, 0)+ d(0, 0, 0, 1).
Okay, any matrix in your space can be written as
You want matrices M1, M2, M3, M4 so that
there are obvious matrices that do that.