So I have been having some difficulties with this problem:

Let $\displaystyle a_0, a_1, ... , a_n be n+1$ distinct real numbers. The Lagrange polynomials are defined by

$\displaystyle
l_j(t):= \frac{(t-a_0) \cdot \cdot \cdot (t-a_{j-1})(t-a_{j+1})\cdot\cdot\cdot(t-a_n)}{(a_j-a_0)\cdot\cdot\cdot(a_j-a_{j-1})(a_j-a_{j+1})\cdot\cdot\cdot(a_j-a_n)}

$

a) Compute the Lagrange Polynomials associated with n=2 and $\displaystyle a_0 = 1, a_1=2, a_2=3$. Evaluate $\displaystyle l_j(a_i)$.

b) Prove that $\displaystyle B = {l_0,l_1,...,l_n}$ is a basis for $\displaystyle R[t]_{\leq n}$.

The first part of a i think i more or less have, here is what I did:


$\displaystyle l_0 = \frac{(x-2)(x-3)}{(1-3)(1-2)} = \frac{x^2-5x+6}{2}$
$\displaystyle l_1 = \frac{(x-1(x-3)}{(2-1)(2-3)} = \frac{x^2-4x+3}{-1}$
$\displaystyle l_2 = \frac{(x-1)(x-2)}{(3-2)(3-1)} = \frac{x^2-3x+2}{2}$


Then I add those together and I get 7. Is this correct atleast?

As for evaluate $\displaystyle l_j(a_i)$, how am i supposed to go about doing this?

For part B I would need to show that the set $\displaystyle B = {l_0,l_1,...,l_n}$ is both linearly independent and spans $\displaystyle R[t]_{\leq n}$. Any suggestions on how to go about doing this?

Thanks for any help you can provide!