Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 5 - Inner Product Spaces - 5.2 Inner Product Spaces - 5.2 Exercises - Page 245: 2

Answer

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) $\langle (u_1,u_2), (u_1, u_2 )\rangle=u_1^2+5u_2^2>0$ and $\langle (u_1,u_2), (u_1, u_2 )\rangle=u_1^2+5u_2^2=0$ if and only if $u_1=0$, $u_2=0$. (2) \begin{align*} \langle (u_1,u_2), (v_1, v_2 )\rangle&=u_1v_1+5u_2v_2\\ &=v_1u_1+5v_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(u_1v_1+5u_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\\ &=(u_1+v_1)w_1+5(u_2+ v_2)w_2\\ &=u_1w_1+v_1w_1+5u_2w_2+ 5v_2w_2\\ &=(u_1w_1+5u_2w_2)+(v_1w_1+5 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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