The task is:
Let ${E := {ax^3+bx^2+cx+d : a, b, c, d ∈ R}}$ be the vector space of all real polynomials of degree at most 3 and let ${F}$ be the subvector space of all real polynomials of degree at most 2. On ${E}$ we consider that for p, q ∈ ${E}$ by.
$${\langle p|q \rangle := \int_{0}^{1}{}p(x)q(x)dx}$$
given scalar product. Calculate the orthogonal projection ${P_F(x^3)}$ and the distance of the monomial ${x^3}$ zu ${F}$.
I am honestly a bit confused and overwhelmed with this. Since I have no clue and am looking for some sort of help to understand this properly. I spent some time googling and using ChatGPT to get around it eventually. The way I found and I tried is:
Finding a basis of F, which could be: $${ax^2+bx+c \Rightarrow x^2+x+1}$$ Then I set: $${P_F(x^3) = ax^2+bx+c}$$ Then I calculated the scalar product. With ${p(x) = x^3}$ and tried finding ${\langle p|P_F(x^3) \rangle}$. This is equal to ${\frac{1}{6}}a + {\frac{1}{5}}b + {\frac{1}{4}}c = 1$. I thought = 1 makes sense since when they both are equal it would be 1. And now I would want to calculate a, b and c. But it feels like this makes no sense and I'd need more equations to do so. I would appreciate a hint or a method to solve this! Thank you.