# Find orthogonal basis for $V^{\perp}$ and find orthogonal projection on $V$.

I have a task in the functional analysis exam that I could not do.

Problem: In Hilbert space $L_2 [0,1]$ (with Lebesgue measure) let:

$$V = \left\{ f \in L_2 [0,1]: \int_{0}^{1} t \cdot \overline{f(t)} = 0 = \int_{0}^{1} (5t^3 + 3t - 2) \cdot \overline f(t) \right\}$$

Find orthogonal basis for $V^{\perp}$ and find orthogonal projection $9t^2 + 2$ on $V$.

My attempt

From definition of $V$ I can write that $V^{\perp} = \hbox{span} \{t, 5t^3 + 3t - 2 \}$. Gram-Shmidt algorithm gives me orthogonal basis contains functions $f(t) = t$ and $f(t) = 5t^3 - 2$.

Now I have to write $9t^2 + 2 = a \cdot t + b \cdot (5t^3 - 2) + \hbox{rest}$. And the rest will by my answer. But I have no idea how can I evaluate it.