## Chapter 6 Exercise C

1. Solution: Suppose $w \in \{v_1, \dots, v_m\}^\perp$. Let $v = \in \operatorname{span}(v_1, \dots, v_m)$. We have that $$ v = a_1 v_1 + \dots a_m v_m $$ for some…

1. Solution: Suppose $w \in \{v_1, \dots, v_m\}^\perp$. Let $v = \in \operatorname{span}(v_1, \dots, v_m)$. We have that $$ v = a_1 v_1 + \dots a_m v_m $$ for some…

1. Solution: (a) One can easily check that each of the four vectors has norm $\sin^2 \theta + \cos^2 \theta$, which equals $1$. Moreover, we have $$ \begin{aligned} \langle (\cos\theta,…

2. Solution: It does not satisfy definiteness. For the function takes $(0,1,0)$, $(0,1,0)$ to $0$, but $(0,1,0)\ne 0$. 4. Solution: (a) Note that $V$ is a real inner product space,…