Question: Equip the real vector space \(V=\mathbb{R}[x]\) ( polynomials with real coefficients) with the inner product

Let

1. Show that the three polynomials \(p_{0}, p_{1}, p_{2}\) are mutually orthogonal in \(V\).
2. Show that the three polynomials \(p_{0}, p_{1}, p_{2}\) span the subspace \(\{f \in V: \operatorname{deg}(f) \leq 2\}\).
3. Let \(g(x)=x^{3}\). Calculate
   
   \[\frac{\left\langle g, p_{0}\right\rangle}{\left\langle p_{0}, p_{0}\right\rangle} p_{0}(x)+\frac{\left\langle g, p_{1}\right\rangle}{\left\langle p_{1}, p_{1}\right\rangle} p_{1}(x)+\frac{\left\langle g, p_{2}\right\rangle}{\left\langle p_{2}, p_{2}\right\rangle} p_{2}(x)\]
4. Find a polynomial of degree three which is orthogonal to the three polynomials \(p_{0}, p_{1}, p_{2}\).

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document