So the question asks: Consider the inner product $⟨f,g⟩=$$ \begin{align} \int_{-1}^{1} f(x)g(x) \ dx &\end{align}$$ $ on $P_2$, the space of all polynomials of degree 2 or less. Find the projection of $f=x^2+6x$ onto the subspace W of $P_2$ spanned by the orthonormal basis $g_1=1/2$ and $g_2= \sqrt{\,1.5 }\ $x. Find $proj_W(f). $
So so far I have:
Because the orthonormal basis are given, then I calculated and got:
||1/2|| = $ \sqrt{\,0.5 }\ $
||$\sqrt{\,1.5 }\ $x|| = 1
then I have an orthogonal basis:
1/ $ \sqrt{\,0.5 }\ $ * 0.5 and 1/1*$\sqrt{\,1.5 }\ $x
So the $proj_W(f) = 2<0.5,f>0.5+1<\sqrt { 1.5 } x,f>\sqrt { 1.5 } x$ =1/3+6x
However, the answer was wrong. So where exactly was wrong?