Give an example of a linear map T:R4→R4T : \mathbf R^4 \to \mathbf R^4 such that

range T=null T\text{range }T = \text{null }T

Consider the right shift operator defined by

T(x1,x2,x3,x4)=(0,0,x1,x2)T(x_1,x_2,x_3,x_4) = (0,0,x_1,x_2)

Clearly T2v=0T^2v = 0, meaning Tv∈null TTv \in \text{null }T. Now suppose Tw=0Tw = 0, clearly w∈range Tw \in \text{range T} since TT spans the nullspace (0,0,x1,x2)(0,0,x_1,x_2).