Let V = P(Q) be the Q-vector space of polynomials of degree at most 5. Let...

Transcribed Image Text:

Let V = P(Q) be the Q-vector space of polynomials of degree at most 5. Let TVV be the linear transformation from V to itself defined as T(f(x)) = xf"(x) 6x f'(x) + 12 f(x). (1) Find the matrix representation A of T with respect to standard ordered bases. (2) Find a basis for the range R(T). (3) Find a basis for the nullspace N(T). (4) Show that there is no nonzero polynomial f(x) = V which is fixed under T, that is, T(f(x)) = f(x) (Hint: consider the system of linear equations (A-16)x = 0). Let V = P(Q) be the Q-vector space of polynomials of degree at most 5. Let TVV be the linear transformation from V to itself defined as T(f(x)) = xf"(x) 6x f'(x) + 12 f(x). (1) Find the matrix representation A of T with respect to standard ordered bases. (2) Find a basis for the range R(T). (3) Find a basis for the nullspace N(T). (4) Show that there is no nonzero polynomial f(x) = V which is fixed under T, that is, T(f(x)) = f(x) (Hint: consider the system of linear equations (A-16)x = 0).