(Solved): Linear Algebra Exercise 6. Prove that \( \Lambda \) represents a projection operator of rank 2. Wit ...

Linear Algebra

Exercise 6. Prove that \( \Lambda \) represents a projection operator of rank 2. Without making any explicit matrix calculations, prove that \( \left(I_{5 \times 5}-\Lambda\right) \) is also a projection operator and determine \( \operatorname{trace}\left(I_{5 \times 5}-\Lambda\right) \) and \( \operatorname{dim}\left(\operatorname{ker}\left(I_{5 \times 5}-\Lambda\right)\right) \). [6 marks] \[ \Lambda=\left(\begin{array}{ccccc} 1 & 1 & 0 & 1 & 1 \\ 3 & 3 & 4 & 3 & 5 \\ 1 & 1 & 2 & 1 & 2 \\ -1 & -1 & -2 & -1 & -2 \\ -2 & -2 & -2 & -2 & -3 \end{array}\right) \]