Consider T LF given by Tz1 z2 z3 2z2 0 5z3 Let w 0 0 1 Com

Consider T L(F) given by T(z₁, z₂, z₃) = (2z₂, 0, 5z₃).

Let w = (0, 0, 1). Compute the list (w, Tw, T²w, T³ w) and then find a linear dependence for this list. Use this to find an eigenvalue for T. Can you find a nonzero eigenvector for this eigenvalue?

Solution

w=(0,0,1)

T(w)=(0,0,5)=5w

T^2(w)=T(T(w))=T(5w)=5T(w)=5^2w

T^3(w)=T(T^2(w))=T(5^2w)=5^2T(w)=5^3w

So an eigenvalue for T is 5 and non zero eigenvector is w as:

T(w)=5w