not to be too nit-picky, but matrices don't "have bases". the column spaces or row spaces of matrices have bases. a basis is a generating set for a vector space, a matrix is just an array of numbers. in fact, a basis is a certain KIND of generating set, a minimal, or linearly independent one (linear dependence means that one or more elements of a set of vectors is redundant, we don't actually need it to recover the whole space. for example, {(1,0),(0,1),(1,1)} is redundant, we don't need (1,1) because we can recover it as (1,0) + (0,1)).

clearly, A does NOT "represent" ALL of R3, it is just some PART of R3. what we want to do is figure out "how big a part"?

one thing we can do, straight-away, is re-write A as:

A = {(a,b,c) in R3: c = a - b} = {(a,b,a-b): a,b in R}.

we can further re-write this as (a,b,a-b) = a(1,0,1) + b(0,1,-1), which suggests that A = span({(1,0,1),(0,1,-1)}).

of course, you haven't yet shown that A is, in fact, a vector space (which you should do), but if it is, then showing {(1,0,1),(0,1,-1)} is a basis

boils down to showing it is a linearly independent set.