Let A= 1 4 1 2

0 2 2 2

1 0 1 2

0 0 2 2,

and let T:C⁴C⁴ be T(v)=Av. It is a fact, which you do not need to prove, that T is nilpotent. Find a cyclic subspace of C⁴ of dimension 3.

Here are some more facts you might want to use (and do not need to prove):

RREF(A)= 1 0 0 1

0 1 0 0

0 0 1 1

0 0 0 0

A²==0 4 4 4

2 4 2 4

0 4 4 4

2 0 2 0

RREF(A²)= 1 0 1 0

0 1 1 1

0 0 0 0

0 0 0 0

You might also want to remember from a prior course that whenever S:CⁿCⁿ is defined as S(v)=Bv for some matrix B, then im(S)=col(B) and ker(S)=null(B).