Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Question

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=5x^2+4$ onto the subspace $V$ spanned by $g(x)=1$ and $h(x)=1$.

Answer 1

We have: $$\langle g,h\rangle = 0,\quad \langle g,g\rangle = \frac{1}{12},\quad \langle h,h\rangle = 1,$$ while: $$ \langle f,g \rangle = \frac{1}{3},\quad \langle f,h \rangle = -\frac{8}{3},$$ hence the projection of $f$ on $\operatorname{Span}(g,h)$ is given by: $$ f^{\perp} =4g-\frac{8}{3}h = 4x-\frac{14}{3}. $$