Isomorphism of P4 and R5 in a given inner product space

[freddyfish]

The isomorphism of ℝ⁵ and P₄ 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 P₄ are uniquely determined by their values at the given values of t_k and each polynomial can be represented as a vector in ℝ⁵ 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 ℝ⁵ 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

 

[chiro]

Hey freddyfish and welcome to the forums.

Can you construct a way to show a co-ordinate transformation from P^5 to R^5?

If you can construct your metric tensor you can show that if the two inner products are equal, then you're done.

 

[freddyfish]

Thanks! I will take a closer look at it, but also, I will soon get my share of tensors, since the autumn offers a course in tensor analysis for me. I'm not "supposed" to know about co-ordinate transformations just yet, but that's what summer vacations are for. To study interesting spaces and messed up co-ordinates, right? :p

 

[chiro]

It's basically taking a lot of stuff you did with going from 2D polar to 2D cartesian and generalizing it in the context of general co-ordinate systems.

Some systems can be really messed up but that depends on your point of view: A lot of the initial examples will be pretty standard like cylindrical, polar, cartesian, parabolic and going between all of these.

Then you look at how you define metrics and inner products and that's the start of generalized co-ordinate system geometry where differential geometry enters the picture.