which problem are you referring to?
I am confused at to why in this solution you have to take two inclusions, doesnt the first half of the solution prove everything. ( ie the bit where you take a linear combination of S).
in general, to show two sets A,B are equal, showing A is a subset of B is not sufficient. B might be bigger.
in your problem, showing that linear combinations of S (elements of span(S)) all have the property that
the sum of the entries are 0, doesn't mean that you have accounted for every single matrix of that form.
you also need to show that if the sum of the 2x2 marix enties are all 0, you can ALWAYS write the matrix as an element of span(S).
so given the matrix A:
[c d], with a + b + c + d = 0, you have to FIND real numbers α1, α2, α3, α4, α5, α6 with:
α1A1 + α2A2 + α3A3 + α4A4 + α5A5 + α6A6 = A. if you can always do this, then
the set of such matrices A is a subset of span(S).
(in point of fact, you only need 3 elements of S. you might suspect this from seeing that if you pick b,c,d at will,
a+b+c+d = 0 forces you to pick -b-c-d for a. that is we can pick a subset of S with the same span,
because S is linearly dependent).