1

my teacher handed out these solutions and for the life of me i can't figure it out

Show that the function $B : V \times V \to \mathbb{R}$, given by $$B(f,g) := \int(1-f(t) g(t))dt$$ and write its Gram matrix with respect to the basis $(1, t, t^2)$.

Matrix:

$$\begin{bmatrix} 2 & 0 & \frac{2}{3}\\ 0 & \frac{2}{3} & 0\\ \frac{2}{3} & 0 & \frac{2}{5}\\ \end{bmatrix}$$

Drinkwater
  • 656
  • 5
  • 13
Dreeww
  • 147
  • First of all, it would be beneficial to learn some LaTeX markup. It makes the question more readable and people are more likely to answer it. I assume you wanted to write the following:
    Show that the function $B : V \times V \to \mathbb{R}$, given by $$B(f,g) := \int(1-f(t) g(t))dt$$ and write its Gram matrix with respect to the basis $(1, t, t^2)$.     – Drinkwater Oct 10 '18 at 13:23
  • Yeah, sorry I tried but it kept putting the wrong syntax – Dreeww Oct 10 '18 at 13:24

1 Answers1

1

This shouldn't be too hard. Assuming from the basis you've given, $V$ is a vector space of polynomials over reals with degree not exceeding 2. Definition of $B$ provided is not entirely correct however, as it contains an indefinite integral. Integral should be definite, but over which subset of $\mathbb{R}$?

As soon as you know that, you can test $B$ for axiomatic properties of an inner product and prove it indeed is an inner product on $V$.

Gram of a basis is also simple. For your basis it would look like this:

$$ \begin{bmatrix} \langle 1, 1 \rangle & \langle 1, t \rangle & \langle 1, t^2 \rangle\\ \langle t, 1 \rangle & \langle t, t \rangle & \langle t, t^2 \rangle\\ \langle t^2, 1 \rangle & \langle t^2, t \rangle & \langle t^2, t^2 \rangle\\ \end{bmatrix} $$ where $\langle f, g\rangle$ means $B(f,g)$. Try computing this and see if the numbers match.

Drinkwater
  • 656
  • 5
  • 13