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Linear transformation, isomorphism and midpoint interpolation

Hi,

a little bit more complex exercise I need help on here. Doc-file is attached.

The first exercise is about linear transformation and isomorphism. The second about some midpoint interpolation. No cheating is involved here. :-) I just don't know how to do this.

Help is highly appreciated!

Re: Linear transformation, isomorphism and midpoint interpolation

For the first part: $\displaystyle T(1)=(1,1,\ldots,1)$ , $\displaystyle T(\cos t)=(\cos \pi/16,\cos 3\pi/16,\ldots,\cos 15\pi/16)$ , ... , transposing we obtain $\displaystyle C$ . Please, try the rest showing some work .