Prove there does not exist a linear map T:R5→R5T : \mathbf R^5 \to \mathbf R^5 such that

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

By 3.22 we must have

dim⁡range T+dim⁡null T=5\dim \text{range }T + \dim \text{null }T = 5

Which is impossible if range T=null T\text{range }T = \text{null }T as it would imply dim⁡range T=5/2\dim \text{range }T = 5/2.