Prove that there does not exist a linear map from F5\mathbf F^5 to F2\mathbf F^2 whose null space equals

{(x1,x2,x3,x4,x5)∈F5:x1=3x2 and x3=x4=x5}\{(x_1,x_2,x_3,x_4,x_5) \in \mathbf F^5 : x_1 = 3x_2 \text{ and } x_3 = x_4 = x_5\}

This nullspace has dimension 2, since we can pick x1,x3x_1,x_3 to determine the others. 3.22 implies dim⁡range T=3\dim \text{range }T = 3, but range T⊆F2\text{range }T \subseteq \mathbf F^2 so it can't have bigger dimension! (see 2.38). Therefor TT cannot exist