# Chapter 5 - Inner Product Spaces - 5.4 Mathematical Models and Least Squares Analysis - 5.4 Exercises - Page 269: 1

see the proof below.

#### Work Step by Step

For any $(u_1,u_2), (v_1, v_2 ), (w_1, w_2) \in {R}^2 , k \in {R}$ (1) $\langle (u_1,u_2), (u_1, u_2 )\rangle=3u_1^2+u_2^2>0$ and $\langle (u_1,u_2), (u_1, u_2 )\rangle=3u_1^2+u_2^2=0$ if and only if $u_1=0$, $u_2=0$. (2) \begin{align*} \langle (u_1,u_2), (v_1, v_2 )\rangle&=3u_1v_1+u_2v_2\\ &=3v_1u_1+v_2u_2\\ &=\langle (v_1, v_2 ),(u_1,u_2)\rangle. \end{align*} (3) \begin{align*} \langle (ku_1,ku_2), (v_1, v_2 )\rangle&=3ku_1v_1+ku_2v_2\\ &=k(3u_1v_1+u_2v_2)\\ &=k\langle (u_1,u_2), (v_1, v_2 )\rangle. \end{align*} (4) \begin{align*} \langle (u_1,u_2)+(v_1, v_2 ), (w_1, w_2)\rangle&=\langle (u_1+v_1, u_2+ v_2 ),(w_1, w_2)\rangle\\ &=3(u_1+v_1)w_1+(u_2+ v_2)w_2\\ &=3u_1w_1+3v_1w_1+u_2w_2+ v_2w_2\\ &=(3u_1w_1+u_2w_2)+(3v_1w_1+ v_2w_2)\\ &=\langle (u_1, u_2 ),(w_1,w_2)\rangle+\langle (v_1, v_2 ),(w_1,w_2)\rangle. \end{align*}

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