# Category: Chapter 7

## Chapter 7 Exercise B

1. Solution: It is true. Consider the standard orthonormal basis $e_1,e_2,e_3$ of $\mb R^3$. Define $T\in \ca L(\mb R^3)$ by the rule:$Te_1=e_1,\quad Te_2=2e_2+e_1,\quad Te_3=3e_3.$Since we have$\langle Te_1,e_2\rangle =\langle e_1,e_2\rangle =0,$$\langle e_1,Te_2\rangle =\langle e_1,e_1+2e_2\rangle =1,$it...

## Chapter 7 Exercise A

1. Solution: By definition, we have \begin{align*}\langle (z_1,\cdots,z_n),T^*(w_1,\cdots,w_n)\rangle=&\langle T(z_1,\cdots,z_n),(w_1,\cdots,w_n) \rangle \\=& z_1w_2+\cdots+z_{n-1}w_{n}=\langle (z_1,\cdots,z_n),(w_2,\cdots,w_n,0)\rangle.\end{align*}Therefore $T^*(w_1,\cdots,w_n)=(w_2,\cdots,w_n,0)$ or $T^*(z_1,\cdots,z_n)=(z_2,\cdots,z_n,0)$. See also Linear Algebra Done Right Solution Manual Chapter 6 Problem 27. 2. Solution: (This solution works for...