## Chapter 8 Exercise D

1. Solution: By Exercise 11 in section 8B the characteristic polynomial is $z^4$ and by 8.46 this is a polynomial multiple of the minimal polynomial. Since $N^3 \neq 0$, it…

1. Solution: By Exercise 11 in section 8B the characteristic polynomial is $z^4$ and by 8.46 this is a polynomial multiple of the minimal polynomial. Since $N^3 \neq 0$, it…

1. Solution. A quick calculation shows that $T^*Tv = ||x||^2\langle v, u \rangle u$ for every $v \in V$. The map $R \in \mathcal{L}(V)$ defined by $$ Rv = \frac{||x||}{||u||}\langle…

1. Suppose $T\in\ca L(U, V)$ and $S\in\ca L(V, W)$ are both invertible linear maps. Prove that $ST\in\ca L(U, W)$ / is invertible and that $(ST)^{-1}=T^{-1}S^{-1}$. Solution: See Linear Algebra Done…