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 - 5.3 Orthonormal Bases: Gram-Schmidt Process - 5.3 Exercises - Page 257: 27


$B'=\{(\frac{3}{5},\frac{4}{5}),(\frac{4}{5},-\frac{3}{5})\}$ is orthonormal basis for $R^2$.

Work Step by Step

Let $B=\{ v_1=(3,4),v_2=(1,0) \}$. Applying the Gram-Schmidt orthonormalization process produces $$w_1=v_1=(3,4)$$ $$w_2=v_2- \frac{v_2\cdot w_1}{w_1\cdot w_1}w_1=(1,0)-\frac{3}{25}(3,4)=(\frac{16}{25},-\frac{12}{25}).$$ Normalizing $w_1$and $w_2$ produces the orthonormal set $$u_1=\frac{w_1}{\|w_1\|}=(\frac{3}{5},\frac{4}{5})$$ $$u_2=\frac{w_2}{\|w_2\|}=\frac{5}{4}(\frac{16}{25},-\frac{12}{25})= (\frac{4}{5},-\frac{3}{5}).$$ Hence, $B'=\{(\frac{3}{5},\frac{4}{5}),(\frac{4}{5},-\frac{3}{5})\}$ is orthonormal basis for $R^2$.
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