?

Linear Algebra

Free Version

Upgrade subject to access all content

Easy

Polynomial Inner Product: Computation

LINALG-RHLFYA

Let $\mathcal{P}_n$ be the set of polynomials with real coefficients and degree as most $n$. This will then form an $n+1$-dimensional real vector space.

We can define an inner product on $\mathcal{P}_2$ by the rule:

$$\langle f,\ g\rangle = f(-1)g(-1) + f(0)g(0) + f(1)g(1)$$

$\ldots$ for all $f,\ g\in \mathcal{P}_2$.

Compute $\langle f,\ g\rangle$ if $f(x) = x^2+x+1$ and $g(x) = 2x^2-x-3$.

A

$0$

B

$\dfrac{3}{2}$

C

$-1$

D

$-9$

E

$-\dfrac{7}{3}$

F

$12$