The isomorphism of ℝ(adsbygoogle = window.adsbygoogle || []).push({}); ^{5}and P_{4}is obvious for the "standard" inner product space.

The following question arise from an example in my course literature for a course in linear algebra. The example itself is not very difficult, but there is a statement without any proof, that if the inner product is determined by:

<p(t), q(t)>=Ʃp(t_{i})q(t_{i})

where t_{k}=-2, -1, 0, 1, 2, respectively, (1≤k≤t)

then the polynomials in P_{4}are uniquely determined by their values at the given values of t_{k}and each polynomial can be represented as a vector in ℝ^{5}where the entries of that the vector (from the top) are the polynomials value for each t_{k}(same order as above).

Intuition confirmes this, but as far as I'm concerned intuition won't prove neither the uniqueness nor why the corresponding vectors in ℝ^{5}is formed from each polynomial's value for each given t_{k}. Can anyone help me to prove this or at least present some idea that might be useful?

I would appreciate it //Freddy

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# Isomorphism of P4 and R5 in a given inner product space

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