Orthogonal complement with non-standard dot product

40 Views Asked by At

When I have a dot product's sequilinear form matrix G in canoninal basis, and I express it in polar basis as

$G'=R^TGR$

how does the orthogonal complement of some subspace $W=<(u)>$ (with canonical basis) change based on the change of basis?

I know to find complement of W with G is to solve

$u.G=0$

If I want to solve it as $u.G'=0$ do I have to express u in the polar basis first?