Advanced linear algebra-basis and linear independence

Question

Problem:

Advanced Linear Algebra : Nilpotent, Vectors, Basis and Linear Independence

Let T: IR³ IR³ , T(x,y,z) = (o,x,2y)

- Show that T is nilpotent of index 3 (that is, T³ = 0 and T² different from 0 )

- Find a vector v E IR^3 s.t T^2(v) different from 0 and show that

B = { v, T(v), T²(v) } is linearly independent (so basis)

- Find A = [ T ] and B = [ T ]

B standard basis

- Find the eigenvalues of T