Let \(V\) be an \(n\)-dimensional vector space over the field \(F\) and let \(T\) be a linear transformation from \(V\) into \(V\) such that the range and null space of \(T\) are identical. Prove that \(n\) is even. (Can you give an example of such a linear transformation \(T\)?)
Solution
Let \[ R=\operatorname{range}T=\ker T. \] Then \[ \operatorname{rank}T=\dim R \] and \[ \operatorname{nullity}T=\dim\ker T=\dim R. \] By rank-nullity, \[ n=\operatorname{rank}T+\operatorname{nullity}T=2\dim R, \] so \(n\) is even.
For example, on \(F^2\) define \[ T(x,y)=(y,0). \] Then \[ \operatorname{range}T=F(1,0)=\ker T. \]