Find the orthogonal projection of the polynomial onto subspace of polynomial

Question

In the space of polynomials of degree at most 2 over the field of complex numbers, the given scalar product: $$(f(x),g(x)) = f(1)\overline{g(1)} + 7f(i)\overline{g(i)} + 7f(-1)\overline{g(-1)}$$ Find the orthogonal projection of the polynomial $$1 + 7ix + x^2$$ onto a subspace of polynomials whose root is 1

So I found a basis of this subspace. It's x-1 and x(x-1). And I don't have any idea what should I do next

Answer 1

Now, apply Gramm-Schmidt to your basis, thereby getting an orthogonal basis $\{e_1,e_2\}$. Then, compute$$\langle1+7ix+x^2,e_1\rangle e_1+\langle1+7ix+x^2,e_2\rangle e_2$$and you'll have the answer to your question.