Hello all..I am really having HARD TIME working the following problem..it took me forever and I am about to give up..I need help or even a hint on how to solve it..it is linear algebra/finding eigenvalues' problem.

Let V = R[X]_2 be the real polynomials of degree at most 2. Let T ( a linear transformation from V to V) be defined as (Tf)(x) = f(x) + f '(x). Find all the eigenvalues of T and their geometric and algebraic multiplicities.

T is a 3 by three matrix, so write it out with respect to the natural basis 1,x,x^2, say (actually, there is a better basis one could choose making the question more obvious, can you see what it is?)

OK..I am not sure if this is correct, but I think I can use the basis B ={ 1, x+1, x^2 + 2x}, or just use the basis in P_2 = { 1, x, x^2}. Then, I computed:

T(1) = 1 + 0 = 1

T(x) = x + 1

T(x^2) = x^2 + 2x

Then, (1)_B = vertical ( 1 0 0) then (x+1)_B = vertical ( 1 1 0) and (x^2 + 2x) = vertical ( 0 2 1 ), so the matrix representation is : with the following 3 columns ( 1 0 0, 1 1 0, 0 2 1), is that right?

Yes, That is correct, so, how many eigenvalues (you can read that off) how many eigne vectors, hence you can work out the algebraic and geometric multiplicities.

 

The better base is {2,2x,x^2}. This is the basis wrt which T is in Jordan Normal Forn.