Linear Algebra/ Finding eigenvalues

[Meital]

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.

[matt grime]

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?)

[Meital]

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?

[matt grime]

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.