Let $V$ be the vector space of $P_2[x]$ of polynomials over $\mathbb{R}$ of degree less than or equal to $2$. Let $L_1$,$L_2$, $L_3$ be the linear functions on $F$ defined by $L_1(f)=f(1)$,$L_2(f)=f(2)$, and $L_3(f)=f(3)$.
Show that the span of the $L_i$'s is a basis for $V^*$ (the dual of $V$). Also, show an explicit basis for $V$ whose dual is $\{L_1,L_2,L_3\}.$
I've been thinking about this for a while. I'm not sure how to show the first part of the second question. Also, I'm curious, is the second question simply using Lagrangean interpolation to find 3 polynomials?
Best Answer
Denote the canonical basis as $(e) = \{t^2, t, 1\}$ and let's try to find a basis $(b)$ such that $\{L_1, L_2, L_3\}$ is its dual basis.
Writing $L_1, L_2, L_3$ in terms of polynomial coefficients, we get
$$L_1(at^2 + bt + c) = a + b + c$$ $$L_2(at^2 + bt + c) = 4a + 2b + c$$ $$L_3(at^2 + bt + c) = 9a + 3b + c$$
Recall that if $I_{(e, b)}$ is a matrix representing the identity with respect to the bases $(b)$ and $(e)$, then the columns of $I_{(e,b)}$ will be the vectors of the basis $(b)$ written in the basis $(e)$.
The rows of its inverse $I_{(b,e)} = I_{(e, b)}^{-1}$ are precisely the matrix representations of the functionals $L_1, L_2, L_3$ in the canonical basis.
Hence, writing $L_i$ as rows$$I_{(b,e)} = \pmatrix{1 & 1 & 1 \\ 4 & 2 & 1 \\ 9 & 3 & 1}$$
and inverting the matrix gives:
$$I_{(e,b)} = \frac12\pmatrix{1 & -2 & 1 \\ -5 & 8 & -3 \\ 6 & -6 & 2}$$
Now we read the vectors of $(b)$ as columns:
$$(b) = \left\{\frac12(t^2 - 5t + 6), \frac12(-2t^2 + 8t - 6), \frac12(t^2 - 3t + 2)\right\}$$