# Consider the function, prove that it's bilinear, symmetric, and positive definite

(Image 1)

For whichever polynomials p(x), q(x) ∈ P≤2. Consider the following polynomials:

p0(x) = 1, p1(x) = x, p2(x) = (3/2 )x² − 1/2 .

(a) Prove that F is bilinear, symmetric, and positive definite

(b) Prove that the family (p0, p1, p2) is an orthogonal basis of P≤2.

Vienrods

28

## Answer

**Answers can only be viewed under the following conditions:**

- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.

Alessandro Iraci

1.7K

The answer is accepted.

Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

- answered
- 600 views
- $8.00

### Related Questions

- Get area of rotated polygon knowing all coordinates and angle.
- Consider the matrix, calculate a basis of the null space and column space
- Linear algebra
- Advice for proving existence claims
- Let $\mathbb{C} ^{2} $ a complex vector space over $\mathbb{C} $ . Find a complex subspace unidimensional $M$ $\subset \mathbb{C} ^{2} $ such that $\mathbb{C} ^{2} \cap M =\left \{ 0 \right \} $
- [Linear Algebra] Spectrum
- Find the null space of the matrix $\begin{pmatrix} 1 & 2 & -1 \\ 3 & -3 & 1 \end{pmatrix}$
- Show that $tr(\sqrt{\sqrt A B \sqrt A})\leq 1$ , where both $A$ and $B$ are positive semidefinite with $tr(A)=tr(B)=1.$