I think what you are saying is this: Let be the subspace of 2 by 2 matrices satisfying
Find a basis for that space.
Okay, go ahead and do the computation: this says
And that reduces to 4 equations: -y= rz, x= rw, -w= -rx, and z= -ry. Since two involve only y and z and two only x and w, that is really two sets of twe equations. Divide the first equation by the last to get -y/z= -z/y which is equivalent to . That means we must have either y= z or y= -z. But what about r? If y= z, then the first equation above becomes -y= rz so r must be -1. If y= -z, then r must be 1.
Divide the second equation the third to get -x/w= -w/x or . We must have either x= w or x= -w. Again, if x= w then r must be -1 and if x= -w, r= 1.
That is, r must be either 1 or -1. Then matrices in are of the form so a basis for this space is the two matrices and .
Matrices in are of the form so that a basis for is given by the two matrices and