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 247: 95


see the proof below.

Work Step by Step

Let $A$ be an $n\times n $ matrix and $u,v\in R^n$, then we have (a) \begin{aligned} \left\langle A^{T} u, v\right\rangle &=\left(A^{T} u\right)^{T} v \\ &=u^{T} A v \\ &=u^{T}(A v) \\ &=\langle u, A v\rangle \end{aligned}. (b) \begin{aligned}\left\langle A^{T} A u, u\right\rangle &=\left(A^{T} A u\right)^{T} u \\ &=u^{T} A^{T} A u \\ &=(A u)^{T} A u \\ &=\langle A u, A u\rangle \\ &=\|A u\|^{2} .\end{aligned}
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