Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 2 - Matrix Algebra - 2.1 Exercises - Page 103: 28

Answer

$u^{T}v = {u}^{T}v$ $(u{v}^{T})^{T}= vu^{T}$

Work Step by Step

The inner product $\mathrm{u}^{T}v$ is a real number (a 1$\times$1 matrix). Its transpose equals itself. By Theorem 3, $(\mathrm{u}^{T}v)=v^{T}(u^{T})^{T}=v^{T}u$ so $\mathrm{u}^{T}v$ = $\mathrm{u}^{T}v.$ The outer product $uv^{T}$ is an $n\times n$ matrix. By Theorem 3, $(\mathrm{u}\mathrm{v}^{T})^{T}=(\mathrm{v}^{T})^{T}\mathrm{u}^{T}= vu^{T}$ Both results are confirmed with the results of the previous exercise.
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