# Linear Functionals and their bases

Let $V=\mathbb{P}_1[x]$ Let $\alpha_{1}(p)=\int_0^1{p(t)}dt$ and $\alpha_{2}(p)=\int_0^2{p(t)}dt$ for each $p \in V$

1. Show $\{\alpha_{1},\alpha_{2}\}$ forms a basis for the dual space
2. Find basis $\{p_1(t)p_2(t)\}$ s.t. $\{\alpha_{1},\alpha_{2}\}$ is the dual basis of $\{p_1(t)p_2(t)\}$
3. Assume $V$ has the inner product defined by $\langle p,q\rangle=\int_0^1{p(t)q(t)}dt$ Find polynomial s.t. $\alpha_2(p)=\langle p,q\rangle$

For 1, since the basis for $V$ is $\{1,x\}$, then evaluating for each function gives non-zero scalars, for the sum of which can only be $0$ iff the functional values have scalar coefficients of zero.

For 2, is this merely a computational question?

For 3, not sure.

EDIT: I think I figured out that the polynomial for $3$ is $t + \frac{1}{2}$

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Is this question referring to the Legendre polynomials? – diimension Nov 12 '12 at 7:42
(3) The correct answer is $q(x)=12x-4$. – user26857 Nov 12 '12 at 10:02
How did you get 12x-4? – Edgar Aroutiounian Nov 12 '12 at 16:52
@Dixie By computation taking $q=ax+b$ and $p=1$, respectively $p=x$ into the formula of $\alpha_2$. – user26857 Nov 13 '12 at 0:39