Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 5 - Inner Product Spaces - 5.3 The Gram-Schmidt Process - Problems - Page 366: 25

Answer

See below

Work Step by Step

We have an orthogonal set of vectors $\{v_1,...v_k\}$ then $=0,\forall i,j \in \{1,...,k\}$ Obtain: $\\ =-\\ =-\sum

\\ =-\sum \\ =-\sum \frac{(x,v_j)}{||v_j||^2}\\ =-\frac{(x,v_i)}{||v_i||^2}\\ =-\frac{(x,v_i)}{||v_i||^2}||v_i||^2\\ =-\\ =0$ Hence, $x-\sum^k_{j=1}P(x,v_j)$ is orthogonal to $v_i,\forall i\in \{1,....,k\}$

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