Give an example of a linear map TT such that dim⁡null T=3\dim \text{null } T = 3 and dim⁡range T=2\dim \text{range } T = 2.

Consider T∈L(R5,R2)T \in \mathcal L(\mathbf R^5, \mathbf R^2) (we pick R5\mathbf R^5 to satisfy 3.22) defined by

T(x1,x2,x3,x4,x5)=(x1,x2)T(x_1,x_2,x_3,x_4,x_5) = (x_1, x_2)

Clearly range T\text{range } T spans R2\mathbf R^2 so dim⁡range T=2\dim \text{range } T = 2. The nullspace is of dimension 3 since we may set x3,x4,x5x_3,x_4,x_5 to anything.