Linear Algebra and Its Applications (5th Edition)

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

Chapter 4 - Vector Spaces - 4.1 Exercises: 14

Answer

$ w\not\in$Span$ S$

Work Step by Step

Let $S=\{v_{1},v_{2},v_{3}\}$. If $ w\in$Span$ S$, then there exist $a,b,c\in \mathbb{R}$ such that $w=av_{1}+bv_{2}+cv_{3}.$ $a, b,$ and $c$ are solutions (if they exist) to a linear system with the augmented matrix $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ -1 & 3 & 6 & | & 7 \end{array}\right]\left\{\begin{array}{l} .\\ .\\ \leftarrow R_{3}+R_{1}. \end{array}\right.$ $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ 0 & 5 & 10 & | & 15 \end{array}\right]\left\{\begin{array}{l} .\\ .\\ \leftarrow (\frac{1}{5}R_{3}). \end{array}\right.$ $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ 0 & 1 & 2 & | & 3 \end{array}\right]\left\{\begin{array}{l} .\\ .\\ \leftarrow R_{3}-R_{2}. \end{array}\right.$ $\left[\begin{array}{lllll} 1 & 2 & 4 & | & 8\\ 0 & 1 & 2 & | & 4\\ 0 & 0 & 0 & | & -1 \end{array}\right]$ (inconsistent, the last row represents the impossible equation $0=-1)$ There are no a,b,c such that $w=av_{1}+bv_{2}+cv_{3}$, so $ w\not\in$Span$ S$
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