Matrix diagonalization and orthogonal projections

Question

Questions:

Linear transformations, Matrix diagonalization and Orthogonal projections

R stands for the field of real numbers. C stands for the field of complex numbers.

1. Let T be a linear transformation from the set P2(R) of all polynomials of degree at most 2 into itself.
                                T: P2(R) --> P2(R), given by

                                                              T(ƒ) = ƒ' - ƒ", ƒ ∈ P2(R),

where ƒ' is the first and ƒ'' is the second derivative of ƒ.

(a) Find the null space N(T) of T. What is the dimension of N(T).

(b) Find the matrix representation [T]_e of T in the basis below

                                    ε = {e₁(x) = 1, e₂(x) = x, e₃(x) = x²}

2. Let a = [²₀;¹_-3]

(a) Find the characteristic polynomial ƒ(t) = det (A - tI).

(b) Find the eigenvalues and corresponding eigen vectors of A.

(c) Is A diagonalizable? Justify your answer by citing a proper theorem.

(d) Find an invertible matrix Q such that D = Q^-1AQ is diagonal.