The subspace { [a b] inM₂ ₓ ₂ |a + d = 0, b + c = 0} and the space of solutions to the homogeneous linear system
[c d]
2x - y - z + w = 0
x + y + z = 0
are isomorphic.
Find a basis for each and use them to give an isomorphism between the spaces. (There are many possible answers here.)