Linear Mathematics | Part IB, 2002

Define the dual space VV^{*} of a finite-dimensional real vector space VV, and explain what is meant by the basis of VV^{*} dual to a given basis of VV. Explain also what is meant by the statement that the second dual VV^{* *} is naturally isomorphic to VV.

Let VnV_{n} denote the space of real polynomials of degree at most nn. Show that, for any real number xx, the function exe_{x} mapping pp to p(x)p(x) is an element of VnV_{n}^{*}. Show also that, if x1,x2,,xn+1x_{1}, x_{2}, \ldots, x_{n+1} are distinct real numbers, then {ex1,ex2,,exn+1}\left\{e_{x_{1}}, e_{x_{2}}, \ldots, e_{x_{n+1}}\right\} is a basis of VnV_{n}^{*}, and find the basis of VnV_{n} dual to it.

Deduce that, for any (n+1)(n+1) distinct points x1,,xn+1x_{1}, \ldots, x_{n+1} of the interval [1,1][-1,1], there exist scalars λ1,,λn+1\lambda_{1}, \ldots, \lambda_{n+1} such that

11p(t)dt=i=1n+1λip(xi)\int_{-1}^{1} p(t) d t=\sum_{i=1}^{n+1} \lambda_{i} p\left(x_{i}\right)

for all pVnp \in V_{n}. For n=4n=4 and (x1,x2,x3,x4,x5)=(1,12,0,12,1)\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=\left(-1,-\frac{1}{2}, 0, \frac{1}{2}, 1\right), find the corresponding scalars λi\lambda_{i}.

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