# vector space

• November 10th 2007, 08:39 AM
smoothman
vector space
any ideas on how to go about conducting these please. i will attempt them once i have a clear idea on how to do this. thanx :)

let V be the vector space of polynomials over C of degree <= 10 and let
"D: V -----> V" be the linear map defined by

D(f) = df/dx

show
(1) D^11=0
(2) deduce 0 is the only eigenvalue of D
(3) find a basis for the generalised eigenspaces v1(0), v2(0), and v3(0).
• November 11th 2007, 12:09 AM
Opalg
Quote:

Originally Posted by smoothman
any ideas on how to go about conducting these please. i will attempt them once i have a clear idea on how to do this. thanx :)

let V be the vector space of polynomials over C of degree <= 10 and let
"D: V -----> V" be the linear map defined by

D(f) = df/dx

show
(1) D^11=0
(2) deduce 0 is the only eigenvalue of D
(3) find a basis for the generalised eigenspaces v1(0), v2(0), and v3(0).

Here are a few hints.
(1) Each time you differentiate a polynomial, you reduce its degree by 1.
(2) If λ is an eigenvalue, with Dx = λx, then $D^{11}x = \lambda^{11}x$.
(3) If I understand this notation correctly, then v1(0) = constants, v2(0) = all linear functions, v3(0) = all quadratic functions. Their bases consist of monomial functions 1, x, x^2.