Let K = <mi mathvariant="double-struck">R and V := <mi mathvariant="double-str

George Bray

George Bray

Answered question

2022-06-06

Let K = R and V := R [ t ] 2 and let F : R [ t ] 2 R [ t ] 2 , f f ( 0 ) + f ( 1 ) ( t + t 2 )
Want to calculate M S S ( F ), the transformation Matrix of F with Basis S, being the standardbasis for the polynomialring S = ( 1 , t , t 2 )
So we have to use the coordinate isomorphism:
M S S ( F ) = ( I S ( F ( 1 ) ) I S ( F ( t ) ) I S ( F ( t 2 ) ) )
Already know the solution, however I have no idea why the following matrix is the transformation Matrix of F.
M S S ( F ) = ( 1 0 0 1 1 1 1 1 1 )
Looking at the transformation matrix we know that F ( 1 ) = 1 + t + t 2 ?

Answer & Explanation

zalitiaf

zalitiaf

Beginner2022-06-07Added 27 answers

Given a polynomial f ( t ) V,
F ( f ( t ) ) = f ( 0 ) + f ( 1 ) ( t + t 2 ) .
So, when f ( t ) = 1, we have f ( 0 ) = f ( 1 ) = 1 and then F ( 1 ) = 1 + 1 ( t + t 2 ) = 1 + t + t 2
Micaela Simon

Micaela Simon

Beginner2022-06-08Added 3 answers

Thanks for the fast reply! I think I almost got it, f ( t ) = t we have f ( 0 ) = f ( 1 ) = t? Could you tell me what happens in this case I think it will help me understand it better. I've never seen anything like this before and to me it just seems a weird way of writing a function.

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