- Let VV be the three-dimensional space of real-valued polynomials of a real variable of degree ≤2≤2. Let T:V→VT:V→V be the translation operator that acts on a polynomial ff as (Tf)(t)(Tf)(t) = f(t+1)f(t+1). (a) Construct the matrices [T]V[T]V of TT with respect to the basis V=1,t,t2V=1,t,t2.

(b) Show that this matrix has only one eigenvalue (the multiplicity of which is 3). Show that the dimension of the corresponding eigenspace is 11 and find a non-zero vector belonging to this space.

(c) Use the above to define the eigenvalue and the eigenspace of TT