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: 3

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