Find T cyclic operator that has exactly N distinct T-invariant subspaces

[toni07]

Let T be a cyclic operator on $R^3$, and let N be the number of distinct T-invariant subspaces. Prove that either N = 4 or N = 6 or N = 8. For each possible value of N, give (with proof) an example of a cyclic operator T which has exactly N distinct T-invariant subspaces.

Am I supposed to assume that T is a linear operator on V, after assuming that V is a finite dimensional vector space? Also, I think I'm supposed to use the dimension of subspaces, but I don't understand how it's supposed to be used.

 

[jvicens]

Yes, you can assume that T is a linear operator on a finite dimensional vector space V. The dimension of subspaces can be used to prove the number of distinct T-invariant subspaces.

First, let's define a T-invariant subspace. A subspace W of V is called T-invariant if T(W) ⊆ W, meaning that the image of W under T is contained in W.

Now, let's consider the possible values of N.

1. N = 4:
If N = 4, then there are four distinct T-invariant subspaces. This means that V can be decomposed into four T-invariant subspaces, denoted as V = W1 ⊕ W2 ⊕ W3 ⊕ W4. In this case, the dimension of V must be at least 4, since each Wi has dimension at least 1. Since T is a cyclic operator, there exists a vector v in V such that {v, T(v), T^2(v)} forms a basis for V. This basis can be used to construct four T-invariant subspaces: span{v}, span{T(v)}, span{T^2(v)}, and span{v, T(v), T^2(v)}. Therefore, an example of a cyclic operator T with N = 4 is the left shift operator on R^3, defined as T(x,y,z) = (y,z,0). It is easy to verify that T is cyclic and has four distinct T-invariant subspaces.

2. N = 6:
If N = 6, then there are six distinct T-invariant subspaces. This means that V can be decomposed into six T-invariant subspaces, denoted as V = W1 ⊕ W2 ⊕ W3 ⊕ W4 ⊕ W5 ⊕ W6. In this case, the dimension of V must be at least 6, since each Wi has dimension at least 1. Similarly to the previous case, we can construct six T-invariant subspaces using the basis {v, T(v), T^2(v), T^3(v), T^4(v), T^5(v)}. Therefore, an example of a cyclic operator T with N = 6 is the rotation operator on R^3, defined as T(x,y,z) = (y,z,x). It is easy to verify that T is cyclic and has six distinct T-invariant