I'm gonna go ahead and blame lack of sleep as the reason to why I can't do this, lol.

Ok, so I have a set of three 2 x 2 Matrices:

|1 1|, |1 0|, |0 1|

|0 0|, |0 1|, |0 1|

I've got to show that this set is a basis of W where W is defined as a Vector space of upper-triangular 2 x 2 real matrices.

So the first thing I need to do is check to see if the matrices are linearly independent. So would I be right in comparing each matrix with the zero matrix and seeing if the only result obtained is the trivial solution?

Of course not. What you must do is to set up the equation the given matrices, and show that the only solution to such a matricial equation is with
If so, can I just substitute the ones for a and b? Or do I have to split the matrices into column vectors and give them standard scalar variables? This is where I get confused, preparing the proof.

The next thing I guess would be to show that they span the space, so doing like I did when checking independence, except comparing them to [x,y].

Easier and simpler, imo: first show the space of upper-triangular matrices has dimension 3 and thus any lin. ind. set of three vectors (i.e., of three upper-triangular matrices) automatically is a basis Tonio
My apologies for not wording this as well as I could have. I'm kinda in a state of confusion, haha.