## Elementary Linear Algebra 7th Edition

For any $(u_1,u_2,u_3), (v_1, v_2,v_3 ), (w_1, w_2,w_3) \in {R}^3 , k \in {R}$ (1) $\langle (u_1,u_2,u_3), (u_1, u_2 ,u_3)\rangle= u_1^2+2u_2^2+u_3^2>0$ and $\langle(u_1,u_2,u_3), (u_1, u_2 ,u_3)\rangle= u_1^2+2u_2^2++u_3^2=0$ if and only if $u_1=0$, $u_2=0$, $u_3=0$. (2) \begin{align*} \langle (u_1,u_2,u_3), (v_1, v_2,v_3 )\rangle&= u_1v_1+2u_2v_2+u_3v_3\\ &= v_1u_1+2v_2u_2+v_3u_3\\ &=\langle (v_1, v_2,v_3 ),(u_1,u_2,u_3)\rangle. \end{align*} (3) \begin{align*} \langle (ku_1,ku_2,ku_3), (v_1, v_2 ,v_3)\rangle &=ku_1v_1+2ku_2v_2+ku_3v_3\\ &=k( u_1v_1+2u_2v_2+u_3v_3)\\ &=k\langle(u_1,u_2,u_3), (v_1, v_2,v_3 )\rangle. \end{align*} (4) \begin{aligned} &\left\langle\left(u_{1}, u_{2}, u_{3}\right)+\left(v_{1}, v_{2}, v_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle =\left\langle\left(u_{1}+v_{1}, u_{2}+v_{2}, u_{3}+v_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle \\ &=\left(u_{1}+v_{1}\right) w_{1}+2\left(u_{2}+v_{2}\right) w_{2}+\left(u_{3}+v_{3}\right) w_{3} \\ &=\left(u_{1} w_{1}+2 u_{2} w_{2}+u_{3} w_{3}\right)+\left(v_{1} w_{1}+2 v_{2} w_{2}+v_{3} w_{3}\right) \\ &=\left\langle\left(u_{1}, u_{2}, u_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle+\left\langle\left(v_{1}, v_{2}, v_{3}\right),\left(w_{1}, w_{2}, w_{3}\right)\right\rangle \end{aligned}