Find Basis for R4 T-Cyclic Subspace Generated by e1

[hitmeoff]

Homework Statement

For each linear operator T on the vector space V, find an ordered basis for the T-Cyclic subspace generated by the vector z.

a) V = R⁴, T(a+b,b-c,a+c,a+d) and z= e₁

Homework Equations

Theorem: Let T be a linear operator on a finite dimensional vector space V, and let W denote the T-cyclic subspace of V generated by a nonzero vector v $$\epsilon$$ V. Let k = dim(w). Then:

a) {v, T(v), T²(v),..., T^k-1(v)} is a basis for W.

The Attempt at a Solution

v= (1,0,0,0), T(v)= (1,0,1,1), T²(v)= T(T(v))= (1,-1,2,2), T³(v)= T(T²(v)) = (0,-3,3,3)

so basis for W = {(1,0,0,0), (1,0,1,1), (1,-1,2,2), (0,-3,3,3)} ?

 

[lanedance]

the transforms look ok, but the theorem assumes you know k the dimisenion of the T-cyclic subspace generated... how do you know it is 4?

note that
-(1,0,1,1) + (1, -1, 2, 2) = (0,-1,1,1)