$\mathbb{C^2}$ treated as a real vector space

Question

As a real vector space the dimension is 4. What is an orthonormal basis for it with respect to either the standard real inner product?. I've tried gram schmidt with the obvious choice of 4 vectors( 0,1),(1,0),(i,0),(0,i) But I keep getting a new set which isn't even orthogonal.can someone help. If I'm treating it as a real vector surely I am right to use the definition of real inner product and bilinear form as opposed to complex and sesquilinear form just because there's complex numbers involved?

Answer 1

Those four vectors are already orthonormal. Note that the real inner product should be defined as the real part of the complex inner product. So for instance, $\langle (1,0),(i,0)\rangle=\operatorname{Re}(1\cdot (-i)+0\cdot 0)=0$.