Linear Algebra and Its Applications (5th Edition)

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

Chapter 1 - Linear Equations in Linear Algebra - 1.3 Exercises - Page 33: 17

Answer

$h=-17$

Work Step by Step

For $b$ to be in the plane spanned by $a_1$ and $a_2$, the following matrix must be consistent: $$ \begin{bmatrix} 1 & -2 & 4 \\ 4 & -3 & 1 \\ -2 & 7 & h \end{bmatrix} $$ To determine this the conditions for consistency, we row reduce. First, add $-4$ times the first row to the second row: $$ \begin{bmatrix} 1 & -2 & 4 \\ 0 & 5 & -15 \\ -2 & 7 & h \end{bmatrix} $$ Then add twice the first row to the third row: $$ \begin{bmatrix} 1 & -2 & 4 \\ 0 & 5 & -15 \\ 0 & 3 & h + 8 \end{bmatrix} $$ And multiply the second row by $3/5$: $$ \begin{bmatrix} 1 & -2 & 4 \\ 0 & 3 & -9 \\ 0 & 3 & h + 8 \end{bmatrix} $$ We can see by inspection that in order for the second and third lines to not reduce to $0=1$, it is necessary that $h+8=-9$. Therefore, it is necessary that $h=-17$ in order for the system to be consistent and thus for $b$ to be in the span of $\{a_1, a_2\}$.
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