Thread: find the n transformation

let such that Compute

obviously for each

as for the problem, compute the matrix associated to the transformation (it's the canonical basis for so it's easy) and then compute the characteristic polynomial.

use the division algorithm and put (1) where is the characteristic polynomial and is the remainder (its degree is one less than the characteristic polynomial), now you gotta compute

first put and differentiate once and evaluate again at differentiate again and evaluate at this will be useful to find

once got those values, put them at (1) and then evaluate the associated matrix to the transformation; by the Cayley - Hamilton theorem, the matrix evaluated in the characteristic polynomial produces the null one so you'll end up (say the matrix is )

finally, explicitly find and multiply it by and we're done!

Originally Posted by Morgan

let such that Compute

a very easy induction on will prove that:

Originally Posted by Morgan

let such that Compute

Write T as a matrix and calculate the first few powers of T. You should see the pattern pretty quickly.

Or you could just calculate them directly from that formula, but I think it is easier to see with the matrix.